Constitutive relations for creep in polycrystals with grain boundary cavitation

Acra meroll. Vol. 32. No. II, pp. 1977-1990, 1984 Printed in Great Britain. All rights resewed

~Pyri&t

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ooo16160/84 53.00+0.00 1984 pergamon Press Ltd

CONSTITUTIVE RELATIONS FOR CREEP IN POLYCRYSTALS WITH GRAIN BOUNDARY CAVITATION Department

V. TVERGAARD of Solid Mechanics, The Technical University of Denmark, Lyngby, Denmark (Received 29 March 1984)

Abstract-Creep damage, by the nucleation and growth of grain boundary cavities in polycrystalline metals at high temperatures, is incorporated in a set of constitutive relations. The cavity growth on a single grain boundary facet is described by a model that accounts for the influence of grain boundary diffusion as well as dislocation creep of the surrounding grains. This model includes both the mechanism of cteep constrained cavitation, where failure by cavity coalescence occurs at relatively small overall strains, and the range of power law creep dominated cavitation. where rupture occurs at large overall strains. The average tractions on grain boundary facets, determined by this model, are employed in expressions for the macroscopic creep strain-rates. which account for the macroscopic creep dilatation resulting from cavity growth. The temperature dependence of creep rates and damage rates is incorporated through the temperature dependence of the material parameters. The application of the constitutive relations is illustrated by numerical results for thick-walled furnace tubes subject to internal pressure and nonuniform temperature. RQlmrcNous avons introduit darts un ensemble de relations constitutives Ie d&g& de &rage, par germination et croissance de cavitts intergran~aires dam les mctaux polycristallins P haute temp6rature. Nous d&rivons la croissance d’une caviti sur une seule facette d’un joint de grains par un mod&e qui rend compte de l’influence de la diffusion intergranulaire, ainsi que du fluage de dislocations des grains avoisinants. Ce modtle inchrt a la fois la cavitation entrah+e par le fluage, oti la rupture par coalescence des cavit&s se produit pour des dCformations totales relativement faibles et la cavitation domin6 par le Auage en loi de puissance od la rupture se produit pour des deformations totales Clev&s. Nous utilisons les tractions moyeunes sur ks facettes des joints de grains, dCtermin6es par cc mod&e, dans les expressions des vitesses de d&formation du fluage macroscopiques; ces expressions rendent compte de la dilatation provenant darts le @age macroscopique de la croissance des cavitts. Nous introduisons la variation des vitesses de fluage et de d&g&t en fonction de la temperature par l’interm&Iiaire de la variation des param&res du mattriau en fonction de la temperature. Nous illustrons l’application des relations constitutives par des &sultats numtiques concemant les tubes de four a paroi &&se soumis a une pression inteme et a une temp&ature. non uniforme. Zuummenf~Die biem Kriechen von polykristallinen Metallen bei hohen Temperaturen durch Bildung und Wachstum von Hohltiumen an Komgrenxen auftretende Schiidigung wird mit einem Satx von Grundgleichungen beschrieben. Das Hohlraumwachstum an einer einxigen Facette einer Komgrenxe wird mit einem Model1 behandelt, welches sowohl Komgrenxdiffusion afs such Versetxungskriechen der benachbarten Khmer urnfagt. Dieses Model1 erfagt sowohl die durch Kriechen eingeschtikte Hohlraumbildung, bei der der Bruch durch Zusammenwachsen der Hohlriiume schon bei reiativ kleinen Gesamtdehnungen auftritt, als such den Be&h der durch Potenxgeaetxkriechen dominierten Hohlraumbildung, bei da der Bruch bei relativ hohen Gesamtdehnungen atitt. Die mit diesem Model1 erhahenen mittleren Werte fur den Zug auf die facetten werden fur Ausdrticke fur die makroskopischen Kriechraten, die such die makroskopische Vohunaufweitung durch die Hohhaumbildung berticksichtigen, benutat. Die TemperaturabhPngigkeit der Kriech- und Schiidigungsraten wird iiber die Tetnperaturabhitngigkeit der Materialparameter erfallt. Die erhaltenen Zusammenhfnge werden als Beispiel auf dickwandige Ofenrohre, die einem inneren Druck und ungleicher Temperaturverteilung unterliegen, angelwendet.

1. INTRODUCTION

Deformations of polycrystalline metals at elevated temperatures occur mainly by dislocation creep of the grains and by grain boundary diffusion. During the creep process microscopic cavities nucleate and grow on the grain boundaries [l, 21. Coalescence of such cavities leads to microcracks, and the final intergranular creep fracture occurs as these micro-cracks link up.

Experimental results show that cavitation occurs mainly on grain boundary facets normal to the maximum principal tensile stress direction [3-S& In cases where diffusion gives the dominant contribution to the growth of cavities, the rate of growth is constrained by the rate of dislocation creep of the surrounding material, as has been noted by Dyson [a]. Such creep constrained diffusive cavitation has been analysed by Rice [7], to detcnnine time and Hutchinson [8] has proposed

1977

the rupture a phenom-

I978

TVERGAARD:

CONSTITUTIVE

enological consecutive law for the steady creep of polycrystalline materials undergoing creep constrained grain boundary cavitation. In this constitutive description [8] all cavitating grain boundary facets are modelled as traction-free micro-cracks, since it was found in [7] that the normal tractions on the cavitating facets are nearly completely relaxed by the diffusive growth process. The constitutive relations proposed in the present paper are an extension of the relations derived by Hutchinson [8] to also cover the transition range, in which there is less or no creep constraint on grain boundary cavitation. The expressions for the macroscopic creep strain-rates are obtained as a modified version of those in [8], accounting for nonzero normal tractions on the micro-cracks. The development of these normal tractions, and the rupture times for grain boundary facets, are found by a simple model developed by Tvergaard [9]. This model is based on results of Needleman and Rice [lo] and Sham and Needleman [ll] for the rate of growth of a single cavity by the coupled influence of diffusion and dislocation creep. In the range of completely creep constrained cavitation the present constitutive description reduces to a combination of the results in [S] and a multi-axial stress version of [I. The number of cavities on a grain boundary facet has been assumed to be constant in previous investigations [7,9]; but here an approximate model of continuous nucleation is incorporated, based on nucleation mechanisms discussed by Dyson [12] and Argon 121. Much work on the analysis of creep deformation and rupture in structural alloys has been based on the phenomenological damage theory of Kachonov [13]. In this continuum damage theory a scalar damage parameter (or several damage parameters), varying from zero in the undamaged material to one at failure, is ued to describe the rupture process [4,14,15]. Functions are assumed for the dependence of the strain-rate and the rate of growth of the damaged parameter on the current stress and damage and various constants in these functions are determined so that the description agrees with experimental results. The influence of varying temperature on damage has usually not been considered in these investigations. The growing damage parameter has some similarity with growing cross-sectional areas of grain boundary cavities; but continuum damage theory does not attempt to describe the actual microchemical creep failure mechanisms in the polycrystalline material. The application of the consecutive relations proposed in the present paper is illustrated by numerical analyses for thick-walled furnace tubes subject to internal pressure and nonuniform distributions of temperature. The necessary temperature dependence of the constitutive law is introduced through the exponential dependence of the creep strain-rate and the grain boundary diffusion parameter on temperature.

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CREEP

2. CONSTITUTIVE

RELATIONS

The expressions for the rate of creep in a material with a certain density of penny-shaped micro-cracks given by Hutchinson [8], are partly based on results of He and Hutchinson [16] for a single crack in a power law creeping material subject to an axisymmetric remote stress state. Hutchinson [8] derives the strain-rates from a potential function of the overall stress, which is constructed by adding contributions from each penny-shaped crack to the potential function for the untracked material. These expressions are obtained for traction-free micro-cracks; but since incompressibility of the matrix material is assumed, the expressions are also valid for a superposed hydrostatic pressure on the external and crack surfaces. This property will be employed here to account for situations, in which cavity growth is not entirely creep constrained, so that there are nonzero tractions on the surfaces of the micro-cracks. The micro-cracks are used to model cavitating grain boundary facets, and all cracks are taken to be normal to the maximum principal tensile stress direction, since creep experiments indicate that only such facets suffer cavitation [3-5]. Therefore, the scope of this constitutive description is limited to cases where the direction of the maximum principal tensile stress is fixed, or at least nearly fIxed. The matrix material is taken to deform by power law creep, so that the effective creep strain-rate t$ in the absence of cracks is given by d7 (~,/a,)“. Here, a, is a reference stress quantity, n is the creep exponent, (*) denotes differentiation with respect to time and & is a temperature dependent reference strain-rate (see also (2.14) and (2.16) at ,the end of this section). The effective Mises stress is u, = (3s,,~~/2)‘~ in terms of the stress deviator sg = tr# - G%i/3, where uji is the Cauchy stress tensor on the embedded deformed coordinates, G, is the metric tensor in the current configuration, and indices range from 1 to 3. Then, with a uniform normal tensile stress a, on the surfaces of the micro-cracks, the modified version of Hutchinson’s [8] expression for the macroscopic creep strain-rate is

Here, the components of the tensor mv and the value of the maximum principal tensile stress S are related by the expression S = dmii’ The factor p reflects the density of cavitating facets, and Hutchinson [8] found p =4R’h(n

+ 1) I +J -“2 n> (

(2.2)

where R is the radius of the penny-shaped cracks, and A is the number of micro-cracks per unit volume.

TVERGAARD:

CONSTITUTIVE

thus, a value of p of the order of unity corresponds to a case where most of the grain boundary facets (approximately) normal to the maximum principal stress are cavitated. The normal stress a, on the surfaces of the microcracks is considered as an internal variable in the constitutive relations, determined by evolution equations based on a simplified analysis of the creep constrained diffusive cavitation of grain boundary facets [9]. These separate analyses for the facets, to be presented later in this section, also give the rate of cavity growth and the time at which open microcracks are formed by coalescence of the cavities. The derivation of equation (2. l), by simpiy adding contributions from each micro-crack to the macroscopic potential function, is based on the assumption of a dilute con~tration of cracks, so that the cracks can be regarded as isolated and noninteracting. Similar expressions for dilutely voided nonlinear materials have been investigated by Duva and Hutchinson [17], who find that the dilute approximation may be inaccurate for void volume fractions as small as 0.01, in materials with n = 5. For various ~n~ntrations of penny-shaped micro-cracks in a material with n = 5 the accuracy of (2.1) can be estimated by comparison with numerical results for periodically spaced cracks in axisymmetric stress fields [9]_In a case corresponding to p = 0.047, according to (2.2), the axial and radial strain-rates predicted by (2.1) are in excellent agreement with the numerical results (within 2%). For more closely spaced cracks, with p = 0.38, the expression (2.1) underestimates the numerically found macroscopic strain-rates by 18% in the axial direction and 8% in the radial direction. Thus, even at values of p as high as about 0.4 the expression (2.1) gives a useful estimate of the strain-rates. For such high crack concentrations improved estimates can be obtained by choosing b slightly above the value given by (2.2). The solutions of He and Hutchinson [16] for a single penny-shaped crack in a power law creeping material and thus also Hutchinson’s [8] constitutive relations, are obtained for small strains. However, here it will be assumed that (2.1) is also a useful approximation under finite strain conditions, and all equations will be presented in the context of a Langrangian formulation of the field equations. Then, 4; is the creep part of the Langrangian strainrate. Temperature changes affect the strains, due to thermal expansion, and temperature also affects the values of the material parameters. The strain-rates resulting from thermal expansion are here taken to be given by rji= G,aF

(2.3)

where a is the linear thermal expansion coefficient and f is the rate of change of the absolute temperature.

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1979

FOR CREEP

The total strain-rate is taken to be the sum of the elastic part, the creep part and the thermal part, r&=rj;+?j$+~& The elastic stress-strain relationship is taken to be of the form gu = Srru rj&and thus the constitutive relations in the presence of creep and temperature changes can be written as 4;ii = ~~~(~*, _ 4,“,- **:)Here, the elastic instantaneous

(2.4)

moduli are taken to be

G&Gil + GaG’G”

+ -& the Jaumann

G~‘G& 1

(2.5)

rate of the Cauchy stress tensor is

g’j = &” + (G&a”+ G’%“)rj,

(2.6)

E is Young’s modulus and v is Poisson’s ratio. It is noted that with (2.5) the influence of the micro-cracks on the elastic part of the strain-rate is neglected. Now, the separate analyses for the facets, to determine the normal stress o, and the rate of cavity growth, wili be discussed. Thecavities are assumed to be uniformly distributed over the facet, with average spacing 26 and radius u, and the diffusion along the void surface is assumed to be sulEciently rapid, relative to the diffusion along the grain boundary, to maintain the quasi-equilibrium spherical-caps void shape (see Fig. 1). For the angle $ a value around 70” is typical. The growth of a single void in the sphericalcaps shape, by combined grain boundary diffusion and dislocation creep, has been studied numericiilly by Needleman and Rice [lo] and Sham and Needleman [I 11. At sulhciently low tensile stresses cavity growth by grain boundary diffusion is dominant.

(cl Fig. 1.(a) Axisymmetric geometry used to study the growth ofonecavity in the sphericalcaps shape. (b) Equally spa& cavities on grain boundary. (c) An isolated, cavitated grain boundary facet in a poIycrystalline material.

1980

TVERGAARD:

CONSTITUTIVE

Then the rate of growth of the cavity volume is obtained by the rigid grains model, early anaiysed by Hull and Rimmer [3] and subsequently modified by various authors, including Needleman and Rice [ 101 who found .

%--(1 -X)(7,

Y1=4xJIn(l/~)-(3-~)(1-~)/2~

(2.7)

Here, a;is the “sintering stress”J is the area fraction of the grain boundary which is cavitated, and 3 = l)# 6, n/&T is the grain boundary diffusion parameter, where & 6, is the boundary diffusivity, C2 is the atomic volume and kT is the energy per atom measure of temperature. Based on approximate results by Budiansky et al. [IS], for the growth of a spherical void in a power law creeping material, the following volumetric growthrate expressions relating to the spherical-caps shape are employed

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FOR CREEP

as suggested by Sham and Needleman [ll] for high triaxialities. Equation (2.10) shows good agreement with numerically determined growth rates, both for high and low triaxialities 19, lo]. From (2.10) the rate of growth of the cavity radius is found as ri = tf/[47cu2 h(JI)]. The simplified model of cavity growth on a grain boundary facet given in [9] is an extension of the analysis by Rice [7]. Based on the results of He and Hutchinson [ 161, also used for the expression (2. l), the average opening rate 6 of a penny-shaped crack in a power law creeping material subject to an axisymmetric stress state is

(2.11) As in equation (2.1) S is the macroscopic principal stress in the direction normal to the crack, a, is the macroscopic Mises stress with corresponding effective creep strain-rate SF, a, is the average normal tensile stress on the crack surfaces, and R is the radius of the crack. The value of the parameter p is given by the asymptotic expression (2.12)

Here, u, and a, are the mean stress and Mises stress, respectively, representing the average stress state in the vicinity of the void, and the constants are given by a,, = 3/2n, /I. = (n - 1) (n + 0.4319)/n2 and cos Jl]fsin JI. For the high h(#)=[(l +cos$)-‘-f triaxiality range, 04~~~> 1, the expression (2.8) was suggested by Sham and Needleman [ll] and the low triaxiality appro~mation was introduced in [9]. The grain boundary diffusion parameter 9 in a case, where the Mises stress and the corresponding effective creep strain-rate are u, and EF, respectively, has been represented by Needleman and Rice [IO] in terms of a parameter L = (5%o,/c,c”3

(2.9)

which serves as a stress and temperature dependent length scale. Thus, for a/L < 0.1 the total volumetric growth-rate of cavities is very well approximated by (2.7), whereas for larger values of a/L the growth rate is higher than predicted by (2.7), due to an increasing influence of didocation creep. The expression for the rate of growth of the cavity volume used in [9] is 3 = V, + V,,

for

which is highly accurate for all n for 1S/a, 1s 2, but inaccurate in the high triaxility range for S/a, larger than about 3 or 4. Due to the uniformity distributed cavities, with volume V and average spacing 26, the average separation between the two adjacent grains is S = V/X&~, and the rate of growth of this separation is $_ 3 _----* xb2

2v6 zb2b

(2.13)

If the number of cavities on a facet is constant, the ratio b/R remains fixed, and thus 6/b = R/R, which is found from the macroscopic strain-rates. In the expression (2.10) for 3 the first term (2.7) is directly specified in terms of the nom& stress o;, to be found but also a,,, and uCin (2.8) are local average stresses that differ from the known macroscopic stress values. In the present paper 0; is taken to he approximately equal to the known macroscopic value, while a,,, is approximated by the macroscopic value multiplied by u,,/S. This gives a reasonable appro~ma~on of numerically determined stress states around cavitating facets [9], both in the diffusion dominated range where a,/S is cfose to zero, in the creep dominated range where u,,/S is close to unity, and in an intermediate range where the time dependence of a,@ is roughly linear between 1 and 0. A slightly different approximation was used in the simplified model in [9], but numerically the results of the two approaches are indistinguishable from one another. It

TVERGAARD:

CONSTITVTIVE RELATIONS FOR CREEP

was shown in [9] that the simplified model gives a very good approximation of the numerically determined cavity growth rates and rupture times. Now, equality of the two expressions (2.11) and (2.13) for 8 determines both the average normal stress a, on the facets and the cavity growth rate ci. An iterative procedure is used, as the equation is nonlinear in an, due to (2.8). It is noted that for small a/L values, where diffusive cavity growth dominates ( pZ negligible relative to VI), the model baaed on equality of (2. I 1).and (2.13) is a multi-axial stress version of that suggested by Rice [7]. In this range a,/S -N0 is a good approximation, so that also (2.1) reduces to the form suggested by Hutchinson [8]. However, for larger values of a/L, where the contribution of dislocation creep to cavity growth becomes significant, the value of uJS approaches unity, and then the contribution of the micro-cracks to the macroscopic strain-rate tends towards zero. For a structural component subject to a history of varying stress and temperature, the value of the parameter a/L will vary too, and thus quite different mechanisms can control the rate of creep and the rate of cavity growth at different times during the creep rupture process. Grain boundary sliding is not accounted for in the constitutive relations discussed here. In some cases sliding can be markedly reduced by introducing alloying elements, which form concentrated solutions or precipitates at the grain boundary [ 19,201. In other cases the shear stresses on the grain boundaries relax so rapidly, by diffusive motion of atoms along the boundaries, that sliding is almost free, and this can significantly affect the stress distribution in the vicinity of a cavitating facet. In the Appendix some relatively simple adjustments of the present constitutive relations are discussed, to incorporate some effect of grain boundary sliding. Both the effective creep strain-rate df, corresponding to an untracked material, and the grain boundary diffusion parameter 9 are strongly dependent on the temperature [l, lo]. In terms of the absolute temperature T the dependence will here be written on the form

1981

are given by

(2.16)

Here, ()” refers to volume (or lattice) diffusion, ( )# refers to boundary diffusion, G is the elastic shear modulus, b is the length of Burger’s vector, k is Boltzmann’s constant and A is a creep constant. Values of the material parameters (2.16) have been given by Needleman and Rice [lo] and Frost and Ashby [21] for a number of common metals. It should be emphasized that the values of these material parameters are usually not known very accurately, and furthermore no values seem to be tabulated for a number of engineering alloys. For a specific material, relevant values of the parametes in (2.14) and (2.15) may be extracted from experimentally measured creep-rates and creep rupture times at various temperatures and stress levels. However, obtaining realistic diffusion parameters from such comparison of experiments and model calculations requires some knowledge of typical cavity spacings and nucleation rates on grain boundary facets. 3. NUCLEATION OF NEW CAVITIES

In the previous section it has been assumed that the cavities on grain boundary facets are present from the beginning, so that growth alone determines the time to cavity coalescence. This has also been the basis of a number of anlayses of grain botmdary cavitation [ 1,7,9]. However, many metals appear to contain rather few cavities in the initial stages of creep, so that the time required for the nucleation of new cavities plays a significant role. Recently Argon [2] and Dyson [12] have discussed various mechanisms of cavity nucleation. It has usually been assumed the nucleation involves the agglomeration of vacancies on stressed interfaces. This requires very high stresses, and it has been assumed that such high stresses could be produced at obstacles, such as particles, during grain boundary sliding; but that mechanism is not supported by more recent studies. Argon [2] suggests that nucleation depends on localization grain boundary sliding, (2.15) whereas Dyson [ 121explains nucleation as a result of localized slip bands in the grains, leading to high stress concentrations, where they intersect particles where 4, %,, a,, and To are reference strain-rate, on the grain boundary. Both mechanisms depend on diffusion parameter, stress and temperature quandislocation creep, which is stochastic process and this tities, respectively, Q, and Qs are activation energies can explain why nucleation is often found to occur and W is the gas constant. It is noted that the continuously during the creep deformation. temperature dependent reference strain-rate Zr in Newly nucleated cavities are extremeIy small, but (2.1) is specified by the factor multiplying (u,/u,,)” in they must exceed a certain critical size in order to (2.14). The expressions (2.14) and (2.15) are those keep growing. The sintering stress in (2.7) is given by given by Needleman and Rice [lo], when $ and .&S,, g,, = 2y, (sin $)/a, where y, is the surface free energy,

1982

TVERGAARD:

CONSTITUTIVE RELATIONS FOR CREEP

and thus according to (2.7) the critical cavity radius is 2% a, ‘u - sin $ . 0”

(3.1)

When a cavity with radius larger than a, is formed by the stochastic processes, it keeps growing, whereas smaller cavities close up again by diffusion. A number of experimental investigations, discussed by Argon [2] and Dyson [12], have shown that the number of cavities is controlled by the strain rather than the time. Some investigations indicate that the number of cavities is proportional with the effective strain, so that dN/ds, is a constant, where N is the current number of cavities on a grain boundary facet and thus N

dN* -de,“’

s=constant. de,

(3.2)

Values of the nucleation rate given by Dyson [12] show large variations in dN/ds, for different steels. As would be expected, a large value of dN/de, tends to be associated with a low ductility (creep rupture at a small strain) and vice-versa. Some observations indicate that the number of cavities reaches a saturation value N_ before rupture, whereas continuous nucleation until fracture is observed in other cases [12]. The experimental measurements of cavity nucleation are very difficult [2,12], partly because the initial critical radii (3.1) may be as small as lo-’ pm. Most cavities observed in creep experiments, by optical or scanning electron microscopy, are detected at radii of, 0.1 pm, or larger. Thus experimental values of the nucleation rate dN/&, include the time required for the growth of cavities to the limit of detection. In order to incorporate nucleation in the material model presented in section 2, the growth of cavities nucleated at different times should be followed separately. However, the diIIir.sive growth is much faster for small voids than for larger voids, and numerical analyses for nonuniform initial cavity sizes on a grain boundary facet indicate that the smaller voids catch up so rapidly that neglecting the size differences may be a good approximation [9]. Thus, we will make use of the simplifying assumptions that all cavities are of equal size (equal to that of the first nucleated void), and that the cavities can be considered equally spaced throughout the nucleation and growth process. Then, by differentiation of the relationship N = (R/6)2, we obtain the expression 6 d 1N ---=b R 2N

(3.3)

to be substituted in (2.13). If N = 0, the average cavity spacing 26 changes only due to the macroscopic strain, and the first term on the right hand side of (3.3) was also specified in section 2. For N > 0, as given by (3.2), the spacing 26 decays due to nucle-

ation as well, possibly limited by a minimum ratio (b/R),, corresponding to a saturation value N,,, for the number of cavities. The approximations leading to the expression (3.3) are expected to be conservative, since (small) growth times are neglected, so that the resulting rupture times should be somewhat underestimated. If the rate of nucleation dN/ds, is very high, and saturation at a maximum cavity number N,, occurs at an early stage of the creep process, a good approximation will be obtained by neglecting the nucleation times, as anticipated in section 2. However, continuous nucleation throughout the creep process will significantly affect the rupture times. As long as the cavity number N is small, the evolution equations in section 2, will give 0,/S N 1, and thus the macroscopic creep strain-rates (2.1) will be approximately equal to those in an untracked material. Subsequently, after some nucleation u,/S will start to decay, provided that a/L is small, so that the constitutive model predicts an increase of the macroscopic strain-rate for increasing damage. In a structural component subject to nonuniform stress and temperature fields the present constitutive model will give different rates of nucleation and cavity growth in different material points; but in a uniform field the model still implies identical cavity growth on the facets. The unavoidable scatter in such behaviour could be included in the model by dividing the facets in a number of groups, where some groups nucleate cavities earlier than others, perhaps due to a higher concentration of particles on the grain boundaries. Then instead of (2.1) we could take the macroscopic creep strain-rates to be of the form

1

(3.4)

where & is the density of the facets belonging to the kth group and the functions Qk [c+, n, (cr”)J have the same form as the parenthesis multiplying p in (2.1). K = 1 equation (3.4) is identical with (2.1), but for K > 2 each of the different normal stress values (031: in a material point has to be calculated separately by the evolution equations discussed in section 2. The progressive appearance of an increasing number of micro-cracks in the final stages of creep could be modelled in terms of (3.4), by different nucleation laws for the different groups of facets. An extension of (2.1) to the form (3.4) relies on the assumption of a dilute concentration of micro-cracks, neglecting the interaction between neighbouring cracks. Then, by the procedure of Hutchinson [S] and Duva and Hutchinson [17], contributions from the different populations of micro-cracks are simply added to the macroscopic potential function, thus leading to the expression (3.4). This extended version of (2.1) will not be employed in the computations in the present paper; but it is noted that taking K > 1 in (3.4) is only a small extra complication in a computational procedure.

TVERGAARD:

CONSTITUTIVE

A further extension in (3.4) would be to represent populations of micro-cracks with different orientations of the normals for different values of k. This would be somewhat analogous to the phenomenological creep damage approach of’ Hayhurst and Storiikers [15], in which a damage parameter is assigned to every plane within the material, assuming that damage is a function of the normal tensile stress history for each plane. 4. COMPUTATIONAL METHOD

A Lagrangian formulation of the field equations will be employed to solve problems in which the constitutive relations of sections 2 and 3 describe the material behaviour. In terms of the displacement components u’ on the reference base vectors the Lagrangian strain tensor is given by qjj

=

%UiJ

+

Uj.‘/.i

+

Ufj

where ( ).i denotes covariant reference coordinate system. ponents ro of the K.irchhoff current base vectors is defined rii=

(4.1)

SJ)

differentiation in the The covariant comstress tensor on the by

GIga”

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FOR CREEP

1983

pressed by a linear interpolation time t and I + At, respectively $=(l

between the rates at

-Qap”+&p’+W

(4.4)

and a Taylor series expansion is used to estimate the value of the rate at time I + At

Here, by application of (2.4). c+.= 3gus,j/(2a,) is expressed in terms of &. Then, by substitution of (4.4), (4.5), (2.6) and the incremental form of (4.2) into (2.4) the constitutive relations can be written on the form fij = LW ,jk,+ iy, Lifi/ =I SW _ cc~j ,# (4.6) where 9%’ =

z {@W _ f (e* @ + ed Ga + ,#G~ J +a”G”)+a’G”}

(4.7) (4.8)

(4.2)

Jwhere G and g are the determinants of the metric tensors G,, and g,, in the current configuration and in the reference configuration, respectively. The requirement of equilibrium is specified in terms of the principle of virtual work. A linear incremental solution procedure is employed, in which the equations governing the stress increments AT” = t” At, the strain increments Aq,, = rj,, At, etc. during the time increment At are obtained by expanding the principle of virtual work about the current state, using (4.1). To lowest order the incremental equation is AT”@,,+ r”du$5~,~

dV = 1

SI Y

AT’&,dS s

- ,~~~~~~,,dV-~T~~u,dS, V

(4.3)

s

where V and S are the volume and surface, respectively, of the body in the reference configuration, and the terms bracketed in (4.3) are included to prevent drifting of the solution away from the true equation equilibrium path. A forward gradient method, analogous to methods suggested by Peirce et al. [22], is used to increase the stable step size. The expression (2.1) is written on the form ?j;= 6: N,,, where 6: = ir (aJr_Q is the effective creep strain-rate corresponding to the uncracked material, and the tensor N,, is the expression bracketed in (2.1). The effective creep strain-rate is ex-

+@‘G,,a~

. (4.10) l The time increment Ar will be chosen’so that the term aicc/aT (f13 At) is negligible compared with gp’). Thus, a small At is used during rapid temperature changes. Independent of the time step, the thermal expansion, represented by the last term in (4.10), plays an important role at temperature changes, Most of the calculations to be discussed have been carried out with 6 = 0.9, which improves the stability significantly. It is noted that the instantaneous moduli in (4.6) are non-symmetric (Lgk’# Lktiq, both due to the last term on the right hand side of (4.7) and due to the difference between Ml and MI(. The last term in (4.7) depends on volume changes, which cannot be neglected for the material containing micro-cracks, where (2. I) includes a significant creep dilatancy. The two tensors M? and MY are identical in the case of an untracked material (p = 0). but otherwise they will generally differ significantly. The form of the constitutive relations (4.6) is analogous with an elastic-plastic constitutive law for a void containing ductile metal at relatively low temperatures [23,24], where plastic dilatation and non normality of the plastic flow rule result in nonsymmetric instantaneous moduli.

1984

TVERGAARD:

CONSTITUTIVE

In an incremental finite element solution for the material model described here the displacement components ai are approximated by polynomials. The constitutive relations (4.6) are employed in the integration points, used for the numerical evaluation of the integrals in (4.3), and after each increment the currrent values of the stress components and the absolute temperature in the integration points are updated. Also the value of the micro-crack concentration p and the orientation of the micro-cracks is taken to be known at each integration point. It is noted that the incremental stiffness matrix, resulting from the equilibrium equation (4.3), is nonsymmetric due to the nonsymmetry of the instantaneous moduli in (4.6). Both the expressions (2.7)-(2.10) for the rate of growth of a single grain boundary cavity and the expression (2.11) for the opening rate of a pennyshaped crack and derived under the assumption of an axisymmetric state of stress. However, in the application of the constitutive relations it will be assumed that these expressions for the growth rates are also reasonable approximations in nonaxisymmetric stress states, provided that a,,, and u, are the mean stress and Mises stress and that the maximum principal tensile stress direction is normal to the grain boundary facets.

5. APPLICATION TO REFORMER-FURNACE TURRS The coustitutive relations will here be applied to analyse the creep rupture of a thick-walled furnace tube. Such tubes have been analysed recently by Jaske et al. [251, who employ a linear life fraction damage rule [26] to estimate the rupture time in a tube subject to unsteady stress and temperature. This linear life fraction rule makes use of rupture times obtained in uniaxial tests at various constant stresses and temperatures. The method does not account for any influence of the actual multi-axial stress state on damage or for increasing stresses with increasing damage, as is incorporated in continuum damage theory [4,13-151. In the present analyses the initial internal and external radii of the tube are donated by A, and B,, respectively, and the corresponding current radii are denoted by A and B. The tube is subject to an internal pressure p and is heated from the outside, so that the absolute temperatures are TA and TB on the inside and outside, respectively. The periods of transient heat conduction, following temperature changes, are very short (of the order of 1 min) relative to the time required for creep relaxation of thermal stresses (perhaps several weeks). Therefore, the radial temperature distribution in the tube is assumed to be that corresponding to steady heat conduction, so that the absolute temperature

RELATIONS

FOR CREEP

7’(r) at the current radius r is given by T(r)=

T,-(TN--

In(rlB) TA)-. In(A IB)

(5.1)

The material parameters employed for the present study are E = lO”N/m*, v = 0.3, a = 16.8. 10-6(K)-‘, n =5, a,=30.106N/m2, &,= 10-6h-‘, T,=1173K, Q,,/W = 6. lo4 K, 9, = lO-29 m5/Nh and Qe/@ = 3.5. lo4 K. These values do not refer to a particular steel, but are generally consistant with the orders of magnitude of the parameters tabulated by Frost and Ashby [lo, 211, and with creep rates for some heat resistant alloys. Furthermore, the grain size is taken to be such that 41, = 50 pm is representative of the initial radius of the grain boundary facets, the angle defining the spherical shape is taken to be $ = 70” and the initial radius and spacing of the cavities is specified by (a/b), = 0.1 and (b/R), = 0.05. For the sintering stress in (2.7) the value us = 0 is assumed to be a suEiciently good approximation. The initial dimensions of the tube are taken to be A0 = 0.045 m and B,, = 0.055 m, at a uniform temperature equal to T,. Fitting material parameters with test results has been attempted for a particular Cr-Ni-Nb steel, applied for catalyst tubes. Taking minimum creep rates, measured in uniaxial tests at various stresses and temperatures, to represent creep, the corresponding values of n, u,,, 4, TO and Qr/a are readily obtained. The remaining parameters 9,, Q&R, (u/b) and (b/R),, which control the rate of damage, are more difficult. It has been possible to find parameter values, so that the stress and temperature dependent rupture times predicted by the constitutive relations of section 2 fall within the 95% confidence limits of the experiments. However, microscopic observations of characteristic cavity spacings and nucleation rates would be necessary to ensure that the parameter values obtained in this fashion are realistic. A cylindrical reference coordinate system is used for the analysis, with axial coordinate xi, radial coordinate x2 and circumferential angle x3. Neglecting temperature variations along the tube, the solutions are uniform in the axial direction, with a constant axial strain-rate determined so that the resultant axial force in the tube balances the axial component of the internal pressure. A plane strain analysis, neglecting the axial strain-rate, would be a good approximation, if the temperature remained constant. However, with temperature variations and thermal expansion the conditions of generalized plane strain incorporated in the present analyses are necessary to get the correct values of the axial stresses. The radial displacements are represented by simple one-dimensional finite elements through the thickness, and in each increment a mixed finite elementRayleigh Ritz solution [27] of (4.3) is used to determine the radial displacement-rates at the nodal points

TVERGMRD:

CONSTITUTIVE RELATIONS FOR CREEP

and the corresponding axial strain-rate. The analyses to be presented have been carried out with 10 linear displacement elements through the thickness, using one central integration point in each element. The principal stress directions remain constant in the tube, and the maximum principal tensile stress is usually the hoop stress, except for shorter periods of time, where thermal gradients can give compressive hoop stresses in part of the tube. Thus, the microcracks with density p are taken to be in the x’-x2-planes, with the macroscopic normal principal stress S = a”G,, in (2.1) and (2.11). In each of the cases to be considered the tube is taken to be stress free initially, at a uniform temperature. An internal pressure p = 3.5. IO6N/m2 is applied at time I = 0, and simultaneously the new temperature distribution is applied. A relatively high concentration, p -0.5, of cavitating facets is assumed. The reference time to be used in the following is defined by I, = u,/(,!?dR, where a, is a Mises stress obtained from the average axial and circumferential stresses, and C,’ is the corresponding effective creep strain-rate at temperature Tw In all cases to be discussed the reference time is t, = 6974 h (0.796 years). In the first case, illustrated in Figs 2 and 3, the temperature remains uniform, TA = T, = To, after that the internal pressure is applied. Figure 2 shows the radial variation of strains and stresses after some creep, at t/tR = 5.95, where the creep strains are about 13 times the initial elastic strains. Fig. 2(a) shows that the circumferential logarithmic strain a1 is largest at the inside of the tube, and the constant axial strain E, is small, indicating that a plane strain assumption would have been useful in this case. The principal true stresses in Fig. 2(b), normalised by the internal

pressure p, show that the hoop stress u, is indeed the largest tensile principal stress. The compressive radial stress varies between the value of the internal pressure at the inside of the tube and zero at the outside. Figure 3 shows the development of the macroscopic hoop stresses S E us, the normal stresses U, on the cavitating grain boundary facets, and in the radius to average half spacing ratio a/b for the cavities. The initial elastic stress distribution has the peak stress at the inside of the tube; but after some creep the peak is at the outside, and subsequently the distribution stays like that, with a slowly increasing stress level due to expansion of the tube. Coalescence of the cavities, a/b = 1, occurs first on the facets near the inside of the tube, at time r/r,, = 61.7. The small values of u,/S throughout the creep process show that cavitation is creep constrained in this case, and it is characteristic that coalescence occurs at a relatively small macroscopic strain, sj = 0.024, in the direction normal to the cavitating facets. This creep constrained cavitation corresponds to a relatively low value of the ratio a/L, which is here around 0.015 at the initial stage. When coalescence of the cavities has taken place on a grain boundary facet, the micro-cracks open up by a mechanism that involves dislocation creep of the grains as well as grain boundary sliding. Numerical studies of this mechanism indicate that the macroscopic creep strain-rates for given stresses are much increased in this final stage of the creep rupture process [28]. To get a rough estimate of the behaviour of the tube beyond the first coalescence, the computation has been continued, still using the expression (2.1) everywhere; but taking p to be 30 times larger after coalescence. Thus p p: 15 and u, = 0 is used for a/b = I, and this the basis of the curves for

f \

3. 2x16 \

k \,

g3

-g2

1i

3. IO

d

a5 x2-A

1985

1

0

Ro-A0 Bo- A0 (al (bl Fig. 2. Pressurized tube with uniform temperature TA= Td = To.(a) Radial variation of logarithmic strains at r/f, = 5.95. (b) Radial variation of true stresses at t/la = 5.95.

TVERGAARD:

1986

CONSTITUTIVE

RELATIONS

FOR CREEP

2 s'+

L *0

a52 1 x -Alz kAO lal

Bo- Acl

(b)

Fig. 3. Development of stresses and damage in pressurized tube with uniform temperature TA= TB= T,. Initial cavities specifkd by (a/b), = 0.1 and (b/R), = 0.05. (a) Macroscopic hoop stress. (b) Normal stress on cavitating facets. (c) Cavity growth.

t/t, = 71.0 in Fig. 3. It is seen that this model gives a ~st~bution of the stresses, so that the rate of cavity growth is speeded up by higher stresses in the region where a/b is still below unity. A further improvement of the material description might be obtained by gradually increasing the value of p after coalescence, so that the stresses in this region would gadfly decay towards zero. It is noted that calculating the rate of cavitation by the model (2.7)~(2.13) is independent of whether or not the expression (2.1) is nsed for the macroscopic

creep strain-rates. The influence of using (2.1) for a specified value of the crack density p is that this affects the stress distribution in a problem with nonuniform stress fields, and since the rate of cavitation depends on the macroscopic stress level, the application of a realistic expression for the macroscopic creep strain-rates is important. The computation illustrated in Figs 2 and 3 has been repeated for p -0, thus neglecting the influence of the micro-cracks on the creep rates and simply taking creep to be incompressible. This different creep law

0.6 -

eoAo

BO-AO Bo-Ao IC) Ib1 Fig. 4. Development of stresses and damage in pressurized tube with non-uniform temperature, T,, = To - 2OK and T, = To+ 20 K. Initial cavities specified by (a/b), = 0.1 and (b/R), = 0.05. (a) Macroscopic hoop stress. (b) Normal stress on cavitating facets. (c) Cavity growth. (at

TVERGAARD:

CONSTITUTIVE

gives slightly smaller hoop stresses near the inside of the tube, which results in a 16% higher time to first coalescence. It is expected that this effect of the p value will be more pronounced in problems with more strongly non-uniform stress distributions. Figure 4 shows results for a tube subject to a temperature gradient, as specified by TA = To - 20 K The internal pressure and T,=T,+20K. p = 3.5.106 N/m* and the crack density p = 0.5 are unchanged relative to the case illustrated in Fig. 3. The thermal expansion gives rise to significant stress gradients, with compressive hoop stresses near the outside of the tube wall, where the temperatures are highest. These elastic stress peaks are relaxed by creep in the initial stages, and no more compressive hoop stresses are left at t/r a =: 0.15. It is seen in Fig. 4(a) that the thermal stresses are completely relaxed at t/t, = 5.75; but in the present case the hoop stresses remain larger near the inner surface of the tube, due to the smaller creep rates at lower temperatures. The high initial values of u,,/S near the inner and outer surfaces, shown in Fig, 4(b), illustrate the influence of the ratio a/L on cavitation. At the high macroscopic stresses near the surfaces a/L is relatively large, so that cavitation is not constrained by creep. On the other hand, very small values of a/L result from the low macroscopic stresses near the center of the tube-wall, which gives creep constrained cavitation with u,,S N 0. Later in the process, when the elastic stress peaks are relaxed by cteep, u,/S is near zero everywhere. It is noted in Fig. 4(c) that the cavities close in the compressive zone; but they rapidly catch up again, when the stresses become positive. Due to the higher temperature at the outside of the tube, failure occurs nearly simultaneously at

1987

RELATIONS FOR CREEP

the inside and outside (at t/t* = 67.5). The larger time to failure than that found in the case of Fig. 3 is a result of the lower temperature on the inside of the tube in Fig. 4. TO model the nucleation of cavities a computation has been carried out for a problem identical with the one described above, except for the initial values (b/R), = 1, (u/b), = 0.005 and the continuous nucleation specified by the value dN/d$= 10’ in (3.2). According to these values the initial number of cavities on a grain boundary facet is assumed to be N = 1, and the initial radius a, of this cavity is assumed to be the same as that used in Fig. 4. Furthermore, a maximum cavity number N,, is assumed, specified by b/R 2 0.05, so that nucleation stops when the number of cavities used in Fig. 4 has been reached. Results for this computation with continuous nucleation are shown in Fig. 5. The development of the macroscopic hoop stresses in Fig. S(a) is practically identical with that in Fig. 4(a). Due to the very small initial values of a/b, the cavitation process is not creep constrained in the initial stages, and here u,/S is near unity. As in the case of Fig. 4 failure occurs nearly simultaneoulsy at the inside and outside of the tube, but the time to failure is here r/t, = 89.0, or 1.32 times that corresponding to Fig. 4. The maximum cavity number N,, has not been reached at failure in the case of Fig. 5. It is noted that a similar computation, starting with a much smaller cavity radius (a/b),= 0.00005, gives nearly identical failure time (r/tR = 89.3), because the diffusive growth rate (2.7) is very high for small cavities subject to a high stress a,,. Computations with continuous nucleation have also been carried out for a higher and for a lower

1

alb 0.8

0.6

0.6 5.75

0.4

0.2 66.4 i.-Il OO

0.5 2

x -5

1

Bo- *o

Bo-Ao

(al

lb1

(cl

Fig. 5. Development of stresses and damage in pressurized tube with nonuniform temperature, T, = To- 20 K and T, = To+ 20 K. Continuous nucleation, with dN/dsC = Iti. (a/b), = 0.005. (b/R, = 1 and (b/R)& = 0.05. (a) Macroscopic hoop stress. (b) Normal stress on cavitating facets. (c) Cavity growth. AM 32,11--H

TVERGAARD:

CONSTITUTIVE

RELATIONS

FOR CREEP l-

1 o/b

%I5 0.8

0.6

0.4 . 68.6 0.2 -

0.2

Q5x2_A ’

L OO

0

o’5 ~2 _A

Bo-Ao

OO

t+

10-7

’

0

Bo- Ao

(b) Fig. 6. Development of stresses and damage in pressurized tube with uniform temperature TA= TB= T,. Diffusion parameter Dotaken to he lo-’ times the value used in previous figures. Initial cavities specified by (o/b), = 0.1 and (b/R), = 0.05. (a) Macroscopic hoop stress. (b) Normal stress on cavitating facets. (c) Cavity growth. rate of nucleation. For dN/de, = l@ in (3.2) the maximum cavity number N,,,,, is reached everywhere at r/f,, 2 15, and failure occurs at r/t, = 69.0. Since this failure time is nearly identical with that corresponding to Fig. 4, it is seen that this type of rapid nucleation up to a maximum cavity number is well approximated by the assumption that the voids are present from the beginning. On the other hand, for a low rate of nucleation, dN/ds, = lo’, the failure time tltR = 183.5 is 2.7 times that corresponding to Fig. 4. A case in which the dislocation creep contribution (2.8) to cavity growth is more dominant, is considered in Fig. 6. Here, the only difference from the computation illustrated in Fig. 3 is that the material is taken to have a smaller diffusion coefficient, D0 = IO-‘* ms/Nh. Thus, the ratio a/L is initially around 0.15 and increses by more than a factor 10 during the cavitation process. In this range uJS remains close to unity, as there is no significant creep contraint on cavitation, and failure (at f/rR = 361) initiates at the outside of the tube, where both the hoop stress a, and the mean stress a,,, are largest. In this case failure occurs at realtively large logarithmic strains, E>- 0.11 at the outside of the tube and E = 0.15 at the inside, as is typical of the creep controlled range with relatively high values of a/L [9]. In contrast to this relatively high creep ductility, the case presented in Figs 2-5 represent rather low creep ductilities, with failure at strains of the order of 0.025. 6. DISCUSSION The constitutive relations proposed here are based on an attempt to describe some of the actual failue

mechanisms in a creeping polycrystalline solid undergoing grain boundary cavitation. Sina the relations rely on detailed models of quasi-equilibrium cavity growth, by the coupled influence of dislocation creep and grain boundary diffusion and on expressions for the rate of opening of penny-shaped cracks in a power law creeping solid, the influence of the multiaxial stress state on the rate of damage enters directly through the mechanisms. Furthermore, the temperature dependence of creep-rates and damage-rates enters the constitutive relations through the temperature dependence of the material parameters. The constitutive relations include the creep constrained cavitation mechanism, as also done by Rice [A and Hutchinson [8]; but in addition the range with no creep constraint on cavitation and the entire transition range between the two are accounted for in the relations. In practice the cavitation process on a given grain boundary facet can be creep constrained in part of the life time and unconstrained or less constrained in other parts. This can be a result of variations in the stress state and temperature or of the rate of cavity nucleation, as is seen in the analyses of thick-walled reformer-furnace tubes. Such variations in the constraints on cavity growth are even more pronounced in actual operating conditions for furnace tubes, where a certain number of shutdowns and upstarts, with corresponding high stresses due to thermal expansion, have to be taken into account. Thus, it seems important for practical applications of the constitutive relations to incorporate a range of mechanisms. Grain boundary sliding gives a noticeable contribution to the deformations in many metals at elevated temperatures, but can be markedly reduced in

TVERGAARD:

CONSTITUTIVE

some cases by concentrated solutions or precipitates at the grain boundary. The constitutive relations in section 2 refer to cases with no (or little) sliding. For materials with almost free grain boundary sliding the type of r&difications discussed in the appendix are expected to give a good appro~mation. The continuum damage theories [4, I3-IS], following the ideas of Kachanov, are an obvious alternative to the type of mechanism based constitutive relations proposed in the present paper. These damage theories have the advantage of representing nucleation laws, statistical distributions of grain sizes, and all other effects, in an average sense, because the damage equations are determined to agree with experiments. However, if the grain size or any other parameter is changed, a new series of experiments seems to be required for this approach. Furthermore, little work seems to have heen done to incorporate unsteady temperatures in the continuum damage theories. Such analyses can be based on linear life fraction damage rules, as has heen done for furnace tubes 1251 subject to unsteady temperatures and stresses; but this approach seems to lack some of the refinements built into the Kachanov type damage theories. All these creep damage analyses neglect the macroscopic creep diiatency resulting from cavity growth, which may have a noteworthy influence on the stress distributions and thus the rupture times. Also the application of the constitutive relations proposed in the present paper requires knowledge of the values of several parameters, inclucing creeprates, diffusion constants, activation energies, nucleation rates etc. Some of these parameters can be determined rather accurately from a few experiments, some can be extracted (appro~mately) from tables of material properties, and the rest will have to be chosen so that experimental rupture times are well reproduced by the. model. If these material parameters, typical cavity spacings, etc. are known with a reasonable accuracy, it shoufd be possible to reIy on the predictions of the constitutive relations over a wide range of stress states and temperatures. In any case, choosing the parameters so that rupture times, measured in uniaxial tests at steady stress and temperature, are well represented, has been found possible. Since the model is based on the actuai failure mechanisms, a better representation of unsteady conditions is expected than that found by a linear life fraction rule.

REFERENCES 1. A. C. F. Cocks and M. F. Ashby. Prog. Mater. Sci. 27, 189 (1982). 2. A. S. Amon, in ¢ Adwnees in Creep mul Fracture of Engi&erfng Materiah and Sfrtictures~(edited by B. Wilt&ire, and D. R. J. Gwen) p. 1, Pinerage Press, Swansea, U.K. (1982). 3. D. Hull and D. -E. F&mer, Phil. Mug. 4, 673 (1959). 4. W. A. Trampczynski, D. R. Hayhurst and F. A. Leckie, J. Mech. Phys. So&& 29, 353 (1981).

RELATIONS

FOR CREEP

1989

5. B. F. Dyson, A. K. Verma and Z. C. Szkopiak, Acta metail. 29, 1573 (1981). 6. B. F. Dyson, Metal Sci. 10, 349 (1976). 7. J. R. Rice, Acta metall. B, 675 (1981). 8. J. W. Hutchinson, Acta metaN. 35 1079 (1983). 9. V. Tvergaard, J. Me&. Phys. So&& 32, (1984). In press. 10. A. Necdleman and J. R. Rice. Acln mcl& 23, 1315 (1980). 11. T.-L. Sham and A. Needleman, Acta mefaN. 31, YlY (1983). 12. B. F. Dyson, Scripta metall. 17, 31 (1983). 13. L. M. K&hanov, &I. A&ad. Nu&. Uk.S.R. Otd. Tekk. Nauk. 8, 26 (t 958). 14. D. R. Hayhurst and F. A. Leckie, Proc. 4th Int. Co!6 Mech. o/ Muter. (edited by J. Carkson and N. G. Ohkon), Vol. 2, p. 1193. Pergqnon Press, Oxford (1984). 15. D. R. Hayhursl and B. %or&kers, Pm. R. Sot. Land. A 349, 369 (1976). 16. M. Y. He and J. W. Hutchin~n, 3. appt. Mech. 48,SM (1981). 17. J. M. Duva and J. W. Hutchinson, Comtituriue Potentials for Dilutely Voided Nonlinear Materials Division of Appt. Sciences, Harvard University (1983). 18. B. Budiansky, J. W. Hutchinson and S. Slut&y, in Mcchanirs of So&&, The Roabey Hilt 6&h Amive~sary Voiume (edited bv H. G. Hookins and M. J. Seweli).,I Vat. 13, ‘Pergamin Press, Oiord (1982). 19. M. F. Ashby, Surf. Sci. 31, 498 (1972). 20. F. A. McClintock and A. S. Argon, Mechanical Behauiour of Materials. Addison Weslev. Readina. MA

(1966): 21. H. J. .Frost and M. F. Ashby, &formation-Mechanism Maps, The Plasticity and Creep of Metals and Ceramics, Pergamon Press, Oxford (1982). 22. D. Peirce, C. F. Shih and A. Needleman Comput. Struct. 18, 875 (1984). 23. A. L. Gurson, J. Engng Mater. Technol. 99, 2 (1977). 24. V. Tvergaard, J. Me&. Phys. Solid 30, 399 (1982). 25. C. E. Jaske, F. A. Simonen lind D. B. Roach, Hydrocarbon Processing pp. 63-68, January (1983). 26. E. L. Robinson, Trans. A.S.M.E. 74, 777 (1952). 28. V. Tvergaard, .I. Mech. Phys. Solid 24 291 (1976). 28. V. Tvergaard, Techn. Univ. of Denmark, DCAMM Report No. 281 (1984). APPENDIX Grain boundary sliding is not incorporated in the material model discussed in section 2, in which the cavitating grain boundary facets are represented as penny-shaped microcracks in otherwise homogeneous material. Here, some relatively simple attempts to include grain boundary sliding in the constitutive relations will be discussed. For a planar, hexagonal array of grains with freely sliding grain boundaries Rice [‘I]has suggested m~fi~tions of his czep constrained cavitation model, to estimate the influence on the rupture times. The average normal stresses on the facets are directly given by the requirement of equilibrium

in the case of these cylindrical graihs with free sliding, and for the case of uniaxial tension Rice I7] fmds that the macroscopic normal stress S in (2.11) should be replaced by 1.5S. This result is easily generalized to a planer, hexogonal array of grains subject to a multi-axial stress state. Furthermore, Rice 171suggests using an effective facet radius equal to R plus &e-pro&ted length of the inclined sliding f&e& i.e. redacina R in (2.11) by 2R. which is equivalent to replacing #3by 2/J in‘(2.li). _ The grains in an actual three dimensional array are more constrained geometrically than in a plane array. An attempt lo model this has been made by the author (not yet published) in teRns of an axisymmetric numerical model, with free sliding on conical grain boundaries adjacent to the cavitating facet. This model does show an increased rate of

1990

TVERGAARD:

CONSTITUTIVE

opening of the facet and an increased normal stress, as suggested by Rice [7]; but also the stresses parallel to the facet are increased in the grains, so that essentially the mean stress and the normal stress are increased without much effect on the effective Mises stress. Guided by these numerical results and the proposals of Rice [7j a set of modified quantities, denoted by ( )*, are introduced S’=S+c,(S-7)

(AI)

T*=T+c,(S-T)

(A2)

p*=cJl.

(A3)

Here, S and 7”are the actual macroscopic principal stresses in the direction normal to the facet and in the directions tangential to the facet, respectively, and c, and c* are constants. ,Then. in terms of the Cauchy stress sensor av, with S =&n,,, the modified macroscopic normal stress, mean stress and Mises stress am S*=(l

+fc,)a”m,,+r”G,,

(A4)

RELATIONS FOR CREEP

u; = 0,.

(A6)

Using the quantities (A3)-(A6) instead of j?, S, u,,, and (I,, respectively, in the cavity growth model (2.7)-(2.13), a modified normal stress o: on the grain facet results too. Comparison with the numerical computations has indicated that the stress levels as well as the rupture times are reasonably well represented by the modified simple model (2.7)-(2.13), if c, = I and c, = 4 are chosen (the original model corresponds to c, = 0 and c, = 0). The modifications in the cavity growth models should be accompanied by the corresponding modifications of the expression (2.1) for the macroscopic creep strain-rates. Thus, with m,j

=

(1 + $ c,) m,, - +,

p+=cfl

(A7) (Ag)

the quantities p, S, u, and my in (2.1) should be replaced by p*, S+, u: and m& respecttvely.