Prediction of the first spinning cylinder test using continuum damage mechanics

Nuclear Engineering and Design 152 (1994) 1-10

ELSEVIER

Nuclear Engineed.ng arid Oesagn

Prediction of the first spinning cylinder test using continuum damage mechanics D.P.G. Lidbury a, A.H. Sherry a, B.A. Bilby b, I.C. Howard b, Z.H. Li b, C. Eripret c a AEA Technology, Reactor Services, Risley, Warrington, Cheshire, WA3 6AT, UK b SIRIUS, University of Sheffield, Department of Mechanical and Process Engineering, Sheffield, $I 3JD, UK c Electrict~ de France, D~partment MTC, Les Renardi~res, BP1, 77250 Moret-sur-Loing, France

Abstract For many years large-scale experiments have been performed world-wide to validate aspects of fracture mechanics methodology. Special emphasis has been given to correlations between small- and large-scale specimen behaviour in quantifying the structural behaviour of pressure vessels, piping and closures. Within this context, the first three spinning cylinder tests, performed by AEA Technology at its Risley Laboratory, addressed the phenomenon of stable crack growth by ductile tearing in contained yield and conditions simulating pressurized thermal shock loading in a PWR reactor pressure vessel. A notable feature of the test data was that the effective resistance to crack growth, as measured in terms of the J R-curve, was appreciably greater than that anticipated from small-scale testing, both at initiation and after small amounts (a few miUimetres) of tearing. In the present paper, two independent finite element analyses of the first-spinning cylinder test (SC 1) are presented and compared. Both involved application of the Rousselier ductile damage theory in an attempt to understand better the transferability of test data from small specimens to structural validation tests. In each instance, the parameters associated with the theory's constitutive equation were calibrated in terms of data from notched-tensile and (or) fracture mechanics tests, metallographic observations and (or) chemical composition. The evolution of ductile damage local to the crack tip during SC 1 was thereby calculated and, together with a crack growth criterion based on the maximisation of opening-mode stress, used as the basis for predicting cylinder R-curves (angular velocity vs. Aa, J integral vs. Aa). Except in the initiation region, the results show the Rousselier model to be capable of predicting correctly the enhancement of tearing toughness of the cylinder relative to that of conventional test specimens, given an appropriate choice of finite element cell size in the region representing the crack tip. As such, they represent a positive step towards achieving the goal to establish continuum damage mechanics as a reliable predictive engineering tool.

1. Introduction During the last decade, several programmes have been mounted validate the fracture mechanics ployed in the structural integrity

large-scale test world-wide to principles emassessment of

L W R pressure vessels. In particular, the spinning cylinder test facility (Clayton, 1985) was designed and constructed to validate the fracture mechanics principles used in U K civil P W R pressure vessel safety cases. To date, six spinning cylinder tests have been conducted. The first three tests were

0029-5493/94/$07.00 © 1994 Elsevier Science S.A. All rights reserved SSDI 0029-5493(94)00797-3

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D.P.G. Lidbury et aL / Nuclear Engineering ana Design 152 (1994) 1 - I 0

aimed at a progressive demonstration of stable crack growth by ductile tearing in contained yield and Conditions simulating pressurized thermal shock loading, Each involved a full length axial defect (fatigue precracked to ao/W= 0.55) in a cylindrical test specimen of modified A508 class 3 pressure vessel steel preheated to a temperature of 290 °C. In the first test ductile crack growth was generated by progressively increasing the rotation speed to simulate pressure loading. In the second test ductile crack growth was generated by thermally shocking the inner surface of the cylinder with water at ambient temperature. In the third test it was generated by combined rotational and thermal shock loading. In each case a notable feature of the test data was that the effective resistance to crack growth, as measured in terms of the J R-curve, was appreciably greater than that anticipated from small-scale (compact specimen) testing, both at initiation and after small amounts (up to a few millimetres) of tearing. This effect, whilst not explainable in terms of the conventional theory of J-controlled growth (McMeeking, 1979), must be ultimately understandable in terms of the variation of crack-tip stresses and strains as a function of geometry and loading configuration, and the response of the material to these variations. On this basis, the transferability of data from standard compact fracture mechanics specimens to spinning cylinder tests may be investigated numerically by simulating the evolution of ductile damage caused by the nucleation and growth of micro-voids in response to crack-tip stress and strain fields. This enables direct predictions to be made of crack initiation and subsequent growth. By combining the results of the numerical simulation with independent Jintegral calculations, J R-curves may be calculated. By performing separate computations for compact specimen and cylinder it is possible to see whether their respective crack growth responses may be reconciled in terms of a transferable constitutive equation representing the ductile crack growth process. The purpose of this paper, then, is to compare two recently published finite element analyses (Bilby, 1992; Eripret, 1991) of the first spinning cylinder test (SC 1), both of which used the above

damage mechanics approach. Each involved application of the Rousselier ductile damage theory (Rousselier, 1987) in an attempt to address key questions regarding the transferability of fracture toughness test data to structures. In both instances, the parameters associated with the theory's constitutive equation were calibrated in terms of data from notched-tensile and (or) compact specimen tests, metaUographic observations and (or) chemical composition. The evolution of ductile damage in response to the local (crack-tip) values of stress and strain during SC 1 was thereby calculated and, together with a crack growth criterion based on the maximization of opening-mode stress, used as the basis for predicting cylinder R-curves (angular velocity vs. Aa, J-integral vs. Aa). The results of these studies are therefore examined to see whether the continuum damage mechanics approach has potential for becoming a reliable predictive tool for the transfer of ductile tearing test results to structures.

2. Outline details of the first spinning cylinder test

The experimental details of spinning cylinder test 1 are described fully in Clayton (1985). Briefly, the general arrangement of the apparatus is shown in Fig. 1, where the central feature is an 8 ton cylindrical test specimen (1.3 m long, 1.4 m OD, 200 mm wall thickness) suspended by a flexible shaft from a single pivoted bearing so that it is free to rotate about the vertical axis. The driving power is provided by a 375 kW d.c. motor mounted on a horizontal pedestal, and is transmitted via a right-angle gearbox with 2:1 step-up ratio (maximum design speed of 3500 rev rain -1 at the rotor). A damping device (not shown) is attached to the bearing pivot to stabilize the rotor against aerodynamically induced precessional motion. The cylinder is suspended in a reinforced underground enclosure for safety containment. Eight 3 kW heaters mounted vertically within the test enclosure provided the necessary thermal energy to raise the temperature of the cylinder to a pretest value of 290 °C. The rotation speed of the cylinder was measured by three independent devices. The primary

D.P.G. Lidbury et al. / Nuclear Engineering and Design 152 (1994) 1-10

3

375 Kw DC Motor .--- 100-way Slip Ring Unit

Junction Box

Right-Angle Gear Box

(2:1 Step Up)

Support Bearing Inertial Damping System Support Shaft Support Disc

riction Dampe__rs

Cylinder Specimen

.,~

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Thermal Shield Hatch -~

l

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Drains

Inspection Chamber--

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Fig. 1. General view of spinning cylinder test rig.

4

D.P.G. Lidbury et al. / Nuclear Engineering and Design 152 (1994) 1-10

speed indication was an analogue tachometer, which also provided the control signal for the motor servo system. The back-up systems were two digital counters, one electromagnetic and the other optical. The primary method employed to measure crack growth during the test was the alternating current potential difference technique (ACPD). Three sets of ACPD probes were situated 25 mm above the bottom of the machined slot in different axial locations. The crack tip was located at the bottom of this slot. The connections for the driving current (0.4 A at 1 kHz) were on opposite sides of the slot so that the current between them passed around the crack tip. The voltage probes were deployed similarly. Back-up measurements of crack growth were obtained from five back face strain gauges welded on the outer surface of the cylinder behind the slot. Additional instrumentation comprised three pairs of clip gauges to monitor changes in the slot gap closely adjacent to the ACPD stations, and an array of thermocouples to measure the variations in cylinder temperature axially, circumferentially and through the thickness. All instrumentation signals were routed through a data logging system that processed and recorded them at preselected frequencies of up to 0.17 Hz. All data were stored to hard disk on line and buffered to a printer. Selected data were also displayed on a visual display unit; in particular, the crack growth signals were further processed by satellite microcomputers to provide a graphical display of growth as a function of speed. In order to generate a J R-curve from the test result the relationship between rotation speed and crack growth was established by a process of calibration. In particular, the temperature-corrected ACPD signal was plotted against the square of the rotation speed and the point at which a pronounced change in slope occurred was identified as the point of tearing initiation. Values of the J-integral corresponding to a particular value of crack growth were obtained by finite element analysis using the ABAQUS (1984) code. Version 4.5 was used, in which values of the J-integral are evaluated by the virtual crack extension method using Parks' (1972) stiffness derivative method. The cylinder was modelled in two-dimensional

plane strain using eight-noded biquadratic quadrilateral elements with reduced integration.

3. The RousseUer continuum damage model The Rousselier continuum damage model (Rousselier, 1987) provides a description of ductile tearing behaviour based on the plastic potential F and the yield criterion F = 0. The resulting constitutive equations, derived using the normality rule (Rousselier, 1987) are (see Appendix A)

F=Fh+Fs

(1)

F h = ~Teq -- R ( p )

(2)

P

Here, Fh denotes the hardening term and Fs denotes the softening (or damage) term. The quantity p is the material density. R(p) is representative of the material true-stress vs. true-strain curve, D is a constant, am is the mean normal stress and trl is related to the material flow stress. The term B(/~) is given by B(~) -

alf0 exp(#) 1 - f o +f0 exp(fl)

(4)

where /~ = In[f( 1 -fo)/fo( 1 - f ) ]

(5)

and fl is the damage variable. The quantities f and f0 are respectively the current and initial values of the void volume fraction. Equivalently,/~ may be calculated from the formula

fl=fDexp(~al) ddPp

(6)

where dPp denotes the equivalent plastic strain rate. The evolution of damage in the above model reflects the competition between material hardening and softening behaviour. A dilatational plasticity represents the growth of voids and leads to softening with increasing deformation. Thus, as loading is increased, the term Fh increases and reflects the increase in crack-tip stresses due to work hardening. With further increases in loading,

D.P.G. Lidbury et al. / Nuclear Engineering and Design 152 (1994) 1-10 00

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Fs increases at the expense of Fh such that the crack opening stress ayy reaches a maximum and thereafter sharply declines (Fig. 2). In conjunction with a finite element model, this effectively allows crack initiation and propagation to be modelled as a progression of discrete steps without recourse to the more usual technique of nodal release (Figs. 3 and 4). Because the above equations do not model the actual linking of voids as the material fails, a crack growth criterion based on stress is invoked. The crack growth criteria used in Eripret and Rousselier (1991) and Bilby (1992) respectively are as follows. (1) When the opening stress reaches a maximum in element n -t- 1, the crack tip is considered

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to move to the boundary of element n, for n = 1, 2, 3 . . . . . Thus initiation occurs when the stress reaches a maximum value in the second element and the crack always moves in steps of L. (2) When the opening stress reaches a maximum at the centroid of element n, the crack tip is considered to move to this location, for n = 1, 2, 3 . . . . . Thus initiation occurs when the stress reaches a maximum in the first element. The crack moves L/2 and thereafter by increments of L. The L values in (1) and (2) above refer to the size of the deformed mesh.

4. Calibration of the RousseHer model

In order to use the Rousselier model, it is necessary to determine the following parameters: the initial void volume fraction fo, the characteristic length 2c describing the ductile fracture process, al and D. In Eripret and Rousselier (1991) and also in Bilby (1992) the value o f f o is equated with the volume fraction of critical inclusions. In both cases this is taken as the volume fraction of MnS inclusions estimated from Franklin's formula as

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Fig. 2. Stress-strain curve showing competition between hardening and softening behaviour. From Eripret and Rousselier (1991).

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2400

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3000

revolutlonslmln

Fig. 3. Normalized hoop stress vs. rotation speed. From Eripret and Rousselier (1991).

For the modified A508-3 steel in question, S=0.012% and M n = 1.32% and so f v = f o = 6.07 x 10 -4.

6

D.P.G. Lidbury et al./Nuclear Engineering and Design 152 (1994) 1-10

There has been, over recent years, a groundswell of criticism of contemporary techniques of damage modelling which use cells identified with the finite element mesh size. This technique, although successful practically in predicting the performance of specimens and structures, as we and others have demonstrated, apparently violates one of the central tenets of numerical analysis of being able to refine the mesh towards a state converged to a prestated accuracy. The current damage mechanics technique does not allow this, because it insists that the smallest mesh refinement is that of the metallurgically defined cell size L. Bilby et al. (1993) have recently been able to address such criticism in research based on the following ideas. Damage is assumed to develop in regions characterized by the fundamental scale length L; this defines a volumetric damage cell. Each cell can contain an arbitrary number of finite elements. Then, in a way whose precise definition depends on the particular model of averaging chosen, the stress and strain histories averaged throughout the elements in a cell are compared with damage theory values until a certain value is attained. At this point, stresses and strains throughout the ceil are replaced by their average values which are then made to follow the damage law. The process is repeated and the number of cells over which averaged stresses and strains are used in the calculations gradually increases as the material damage spreads. In this manner a procedure independent of mesh state (but still dependent on mesh size) is obtained. To date, this process has been attempted for the initiation stage only of SC 1 (Bilby, 1993) and confirms that a similar initiation toughness (independent of crack size) is obtained in the cylinder and the small compact tension specimen. Notwithstanding criticism of metallurgically defined finite elements, the view is taken in Eripret and Rousselier (1991) and Bilby (1992) that the selection of a particular mesh size represents a process of averaging over an appropriate damage cell relevant to the failure mechanism under discussion. Consequently, the mesh size L is equated with the characteristic length 2c describing the ductile fracture process. This in turn equates with the spacing of the MnS particles controlling the failure

process. In Eripret and Rousselier (1991) the value of 2c is estimated to be 550 tim, based on data published in Lacey and Leckenby (1989). This value was therefore used for the finite element mesh size in modelling ductile damage. In Bllby (1992), the best estimate of 2c is 250 0m, based on more detailed metallographic evidence than that available in Lacey and Leckenby (1989). However, for the purpose of assessing the sensitivity of predictions to the value of 2~, finite element mesh sizes of 500, 250 and 125 Ixm were used in Bilby (1992) in modelling the ductile damage process. Subsequent further study suggested that a size less than 250 ~tm might well have been chosen. Another potential concern of the present analyses is that the position of the crack tip at any stage is notional, in that it is marked only by the decline in stress behind it as the damage increases rapidly there. Bilby et al. (1993) have repeated the calculations using a procedure which immediately relaxes the stresses behind the tip to zero after each advance of the crack. After appropriate modification of the criterion for crack advance (which models the termination by actual void coalescence of the dilatational softening provided by the damage theory), similar predictions are obtained. The determination of a~ is generally made via mechanical testing of axisymmetric notched-tension specimens. However, data from such tests were not available at the time Eripret and Rousselier

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D.P.G. Lidbury et al. / Nuclear Engineering and Design 152 (1994) 1-10 80

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8

D.P.G. Lidbury et al./Nuclear Engineering and Design 152 (1994) 1-10

(1991) was published, and so the authors of that paper calibrated Eq. (3) in terms of al with reference to the J R-curve data for 35 mm thick side grooved compact specimens presented in Lacey and Leckenby (1989). The test temperature was 290 °C, corresponding to the temperature at which SC 1 was carried out. A value of al = 350 MPa with fo =fv = 6.07 x 10-4 and L = 550 ~tm gave the best overall prediction of this data, Fig. 5. In Bilby (1992), values of trl for a temperature of 290 °C were determined (D = 2x/~) to be 443, 516 and 571 MPa for L = 500, 250 and 125 ~tm respectively. These values represent a compromise resulting from predicted curves "tuned" to fit not only AE10, AE4 and AE2 notched tensile results 1 but also the results from 35 mm thick side grooved compact specimens, Figs. 6 and 7.

5. Comparison of predictions The finite element analyses in Bilby (1992) and Eriprct and Rousselier (1991) reflect the same set of relevant dimensions for the test cylinder. However, the analysis of Eripret and Rousselier (1991) using the ALIBABA code did not model centrifugal loading; instead, the cylinder was loaded by an internal pressure that would produce in a linear elastic material the same average hoop stress as that in an uncracked rotating cylinder. The numerical simulation of the behaviour of the cracked cylinder thus reflected internal pressure loading; predicted values of crack growth and the J-integral were correlated in terms of the equivalent rotation speed. In the TOMECH code calculations in Bilby (1992), a distribution of body forces was applied to the finite element nodes to simulate the centrifugal loading due to the rotation of the cylinder. Values of the J-integral were obtained by the virtual crack extension method

using an area integral and an interpolation function. No significant differences were found in comparison with corresponding values of the conventional J-integral; moreover, for the particular type of body force loading applied during SC 1, the latter was found to be path independent to within a few per cent, even where the integration path passed through plastic regions, provided the path was not too close to the crack tip. Predictions of crack growth are also reported in Bilby (1992) as a function of rotation speed. These involve the use of the J-integral in the compact specimen calculations only, where values conform very closely to values using the parameter obtained by standard measurements of load and load-line displacement. The predictions of crack growth as a function of rotation speed are thus independent of any complications that may result from the use of the J-integral in relation to body force loading. The J R-curves derived for SC 1 are shown in Figs. 8 and 9. These figures relate to Eripret and Rousselier (1991) and Bilby (1992) respectively. In addition, Fig. 10 shows the predicted and experimental curves from Bilby (1992) for rotation speed vs. crack advance. An important point to note is that in Bilby (1992) the predictions of rotation speed vs. crack advance were made "blind", with the best estimate being for L = 250 ~tm. In all cases the predictions confirm that the cylinder's resistance to ductile tearing is

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1 Three sets of standard axisymmctrically notched bar specimens were tested; all had a bar diameter of 18 mm and a minimum diameter o f 10 mm. The notations AE10, AE4 and AE2 denote notch radii of I0, 4 and 2 m m respectively. Unfortunately only load vs. axial displacement data were available on these notched bars. Fig. 6 shows data for AE10 specimens only.

o o o

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Measurement Station 1 Measurement Station 2 Measurement Station 3 Predict ion

0

0.5

1

1.5

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2.5

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Crack growth, m m

Fig. 8. Predicted and experimental J R-curves for the first spinning cylinder test. From Eripret and Rousselier (1991).

D.P.G. Lidbury et al. / Nuclear Engineering and Design 152 (1994) 1-I0 0.6

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dieted. These results suggest the possibility that initiation in the cylinder may occur at a similar J-integral level to that found in the compact specimens, notwithstanding the slope of the J R-curve being greater in the cylinder. This point is currently the subject of an ongoing study involving quantitative metaUography including detailed post-test measurements of stretch-zone width in both the test cylinder and compact specimens. Lastly, it is noted that in both Bilby (1992) and Eripret and Rousselier (1991) comparisons are made of the fields ahead of the crack tip in the cylinder and the compact specimen at different stages of crack advance using damage theory. In both cases a higher value of the ratio trm/treq of the mean normal stress to the equivalent stress, is reported in the compact specimen compared with the cylinder, albeit after crack initiation in the case of Bilby (1992). Whilst this is consistent with the observation of a higher resistance curve slope in the cylinder, the full explanation of this effect again remains the subject of an ongoing study.

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vs. crack advance in the first spinning cylinder test. F r o m Bilby (1992).

significantly greater than that measured on 35 mm thick compact specimens. Overall, the predictions represent a significant improvement compared with previous predictions based on standard fracture mechanics techniques and small specimen J R-curve data (Lacey, 1989). The best predictions are for mesh sizes greater than or equal to 250 ~tm. In Fig. 8 prediction of the initiation of ductile tearing and the slope of the tearing resistance curve is in good agreement with the experimental measurements. However, there is an underprediction of the overall extent of ductile tearing. In Figs. 9 and 10 there is a deviation from experiment in the region of crack initiation, although the slope of the tearing resistance curve and the total crack extension is accurately pre-

Using the standard Rousselier ductile damage model, comparative predictions were made of crack growth in the first spinning cylinder test carried out by AEA Technology at its Risley Laboratory. The following conclusions may be drawn. (1) Two independent analyses have correctly predicted the post-initiation enhancement in toughness of the cylinder relative to that of standard small-scale fracture toughness specimens. This is a significant improvement compared with previous predictions based on standard fracture mechanics techniques and small-speciman J Rcurve data. (2) The accuracy of predictions is most sensitive to selection of the finite element mesh size L to represent the process of averaging over a damage zone relevant to the failure mechanism under consideration. In the present case this has meant equating L with a characteristic length 2c representative of some average spacing between dominant MnS inclusions.

D.P.G. Lidbury et aL/ Nuclear Engineering and Design 152 (1994) 1-10

10

(3) The potential for models based on continuum damage mechanics to address more complex materials and structural circumstances requires further development and validation. However, the present results represent a positive step towards achieving the goal to establish damage mechanics as a reliable predictive engineering tool.

p al

Appendix A: Nomenclature

O'eq O"m

2¢

O'yy

a/W ao/W Aa D f

fo fv

crack length to width ratio initial crack length to width ratio crack growth increment equivalent plastic strain rate dimensionless constant (D = 2x/~ ) current void volume fraction initial void volume fraction volume fraction of critical inclusions

(fv =f0) F

Fh J L

Mn(%) R(p)

s(%) SC 1 ALIBABA TOMECH

plastic potential hardening component of F softening (or damage) component of F J-integral finite element mesh size selected to represent the process of averaging over the damage zone modelled manganese content (wt.%) term representative of the material true-stress vs. true-strain curve sulphur content (wt.%) spinning cylinder test 1 finite element code finite element code

Greek letters damage variable expressed as a func-

tion of the current and initial values of the void volume fraction parameter representing some average spacing between dominant MnS inclusions material density material constant related to flow stress equivalent stress mean normal stress mode I crack opening stress

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