Experimental and theoretical evaluation of a high-accuracy uni-axial creep testpiece with slit extensometer ridges

Pergamon

Int J. Mech. Sci. Vol.36, No. 8, pp. 751-769,1994

Copyright~ 1994ElsevierScicnc~Ltd Printedin Great Britain.All rights reserved 0020-7403/94 $7.00+ 0.00

0020-7403(94) E0018- E

EXPERIMENTAL AND THEORETICAL EVALUATION OF A HIGH-ACCURACY UNI-AXIAL CREEP TESTPIECE WITH SLIT EXTENSOMETER RIDGES Z. L. KOWALEWSKI,* J. LIN and D. R. HAYHURST Department of Mechanical Engineering, UMIST, P.O. Box 88, Manchester, M60 1QD, U.K. (Received 13 Auoust 1993; and in revised form 14 January 1994)

Abstract--The paper evaluates a previous uni-axial creep testpiece design which has slitted extensometer ridges to relieve the constraint to axial deformation provided by conventional, or unslitted, ridges. The results of experiments are reported on slitted and unslitted ridged testpieees; and they are used together with theoretical studies to evaluate the effectiveness of the slitted ridged testpiece design for different specimen gauge lengths. Firstly, the test results are used to derive true constitutive equations (i.e. those equations for testpieces without ridges), these are then used to predict the response of the majority of the testpieces, using the creep Continuum Damage Mechanics (CDM) Finite Element solver DAMAGE XX. The creep rupture mechanisms, lifetimes and accuracy of the measured creep strains have been determined theoretically for testpieces with unslitted and slitted extensometer ridges and gauge lengths of 50, 30 and 10 ram. Close agreement has been obtained between the experimental and theoretical results for the creep failure mode and lifetimes. It has been shown how the accuracy of the creep behaviour measured using short gauge length testpieces (10 mm), of the type used in tension-compression testing, may be greatly improved by using slitted extensometer ridges.

1. I N T R O D U C T I O N

The measurement of uni-axial creep strain in laboratory tests is frequently carried out using cylindrical bar testpieces on which ridges have been machined to identify the gauge length over which strain is to be measured. Mechanical extensometers are fitted to these ridges which are used to transfer the displacements occurring during creep to a location outside of the high temperature furnace where low cost transducers can accurately measure displacements at ambient temperatures. The measured displacements are then used to compute the variation of strain with time. Alternative techniques include: the measurement of diametral strain using an extensometer and a similar transducer, and the use of high temperature strain gauges which involve welding a substrate to the surface of the testpiece. The former diameteral strain measurement technique requires a hole to be cut in the furnace wall to allow the extensometer rods to couple the testpiece to the external transducer. The hole tends to disturb the specimen temperature gradient, and the technique suffers from the disadvantage that incompressibility of the material has to be assumed in order to calculate axial strains. The latter strain gauge measurement technique suffers from the disadvantages that the gauges only tolerate a maximum strain of typically 5%, they are extremely expensive, not reusable, and the mechanical connection of the substrate to the testpiece can frequently present difficulties. The preferred means of high temperature strain measurement is therefore to use ridged extensometer testpieces and axial extensometers. This type of testpiece and extensometer will be studied within this paper. A previous theoretical study carried out by Linet al. [1] has shown that the uni-axial strain measured in tensile testpieces using ridged specimens and extensometers does not agree with the true strains in the parallel section of the testpiece. The error levels have been shown by Lin et al. [2-1 to be dependent upon the applied stress level, but principally upon the size of the gauge length. For Nickel superailoy specimens, of diameter 7.65 mm, with

* British Council Visiting Researcher at UMIST from October 1992 to July 1993 from the Institute of Fundamental'Technological Research P.A.S., Swietokrzyska 21, 00-049 Warsaw, Poland. 751

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gauge lengths of 51 mm, errors have been predicted in excess of 10%. For gauge lengths of 10 mm the error increases to typically 30%. These figures are for the low stress level of 118 M Pa, which can be expected to double for the stress level of 250 M Pa. Whilst gauge lengths of 10ram are not frequently used in creep testing, they are however used in combined cyclic plasticity and creep testing. The reason for these high errors has been shown by L i n e t al. [2] to be due to the circumferential reinforcement of the testpiece provided by the extensometer ridges. The ridge perturbs the stress, strain and damage fields above and below the ridge, with the perturbations extending a distance of typically 1.5 x the diameter of thc parallel sided region of the testpiece. The circumferential stress generated in the extensometer ridge is predominantly compressive and Lin et al. [2] have shown how these stresses may be relieved by the introduction of slits into the ridges; and how the errors in measured creep strains can, as a consequence, be reduced by a factor of typically two. The same circumferential reinforcement effect has been demonstrated experimentally by Ohashi et al. [3] m tubular specimens subjected to internal pressure. They tested a range of different specimens in which the geometry of the extensometer ridge was varied and shown to strongly affect the measured strains. Since the results due to Linet al. [ 1, 2] are theoretical, there is a need to investigate them further, and to verify them experimentally; this is the major aim of the research reported here. The results of experiments are reported which have been carried out on unslitted and slitted extensometer ridged testpieces with different gauge lengths. The experimental results are compared with those determined theoretically from a knowledge of the uni-axial constitutive equations, and the multi-axial rupture criterion of the material. A precipitation hardened aluminium magnesium-silicon alloy* has been selected for the experimental investigation and tests are reported which have been carried out at 150 + 0.YC. The experimental results obtained on testpieces with unslitted extensometer ridges are used tt~ determine the true constitutive equations for the material. The constitutive equations model primary creep, ageing and creep constrained cavitation as reported by Kowalewski et ~l/. [4]; and, they are used here in a Continuum Damage Mechanics Finite Element analysis. performed by the solver D A M A G E XX [5, 6], to predict the measured creep curves of the slitted extensometer ridged testpieces. These predictions are then compared with the experimental results. Conclusions are made on the effectiveness of the slit extensometer ridged testpiece design at achieving high accuracy measurements of uni-axial creep strains. 2. E X P E R I M E N T A L

PROGRAMME

2.1. Material selection

The aluminium alloy was selected because of its availability and for the existence of results previously obtained on this material by Kowalewski et al. [4] for a different sample of the material, but prepared from the same material cast. It was convenient to test the material at a lower temperature of 150 + 0.Y'C; and, to select the material for its ease of machinability of the slitted extensometer ridges. 2.2. Testpiece manufacture The basic testpieces (Fig. I a) were machined using a DNC Lathe to create the cylindrical specimen form, and a vertical machining centre was used to machine the end flats and loading pin holes. The extensometer ridges were slitted using a dressed circular slitting saw mounted with its arbor located vertically in the machining centre, whilst the axis of the testpiece was mounted horizontally in a dividing head. An engineering drawing of the slitted ridged uni-axial creep testpiece is shown in Fig. lb. Twenty-four slits were machined into each extensometer ridge to reduce the constraint to deformation in the hoop direction, in this way the slitted extensometer ridges were subjected mainly to conditions of plane stress. A finished machined extensometer ridge is shown in Fig. 2. In addition to relieving the hoop stresses, the constraint to axial deformation provided by the extensometer ridges was also

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weakened by the introduction of the slits, since ca 2,'3 of the extensometer ridge matermi was removed. 2.3. Experimental technique There are three factors which are known to significantly influence the accuracy and repeatability of creep tests. They are: material variation; temperature control: and the level of superimposed bending stress. In this work, material variation has been minimised b2, cutting all specimens from a single block of material of dimensions 120 × 120 × 150 ram. This block of material has been itself machined from a large isotropically forged billet of material. The effects due to temperature variation during the test have been reduced by controlling the temperature to 150 + 0.5C; this is equivalent to a percentage change in absolute temperature (°K) of 0.24%. The work of Hayhurst [7] has shown that this produces errors in rupture time of ca 7%. To reduce the temperature control limits to better than +_ 0.5 C would not be practical with the test machinery available. The effects due to superimposed bending stresses have been studied by Hayhurst [7] who has shown that the specimen percentage bending determined from {El -- g2)

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where ~:~ and e2 are uni-axial surface strains measured at diametrically opposed surface points which should not exceed 6%. If these conditions are met, and assuming that specimen deformation takes place to reduce this initial level of bending, then for the aluminium material tested the effect on lifetime would be at worst a reduction of ca 10%. To achieve initial bending levels below 6% the universal block specimen gripping system discussed by Hayhurst [7] has been used. In these ways the effects of material variability, temperature variation and superimposed specimen bending stress have all been reduced. The validity of each test has been assessed by comparison of the experimental results with theoretical curves, and with other tests carried out at the same stress and temperature levels. but sometimes with a different specimen gauge length. The shape and levels of strain rate in the primary and secondary regions of the creep curve, together with lifetimes, have been assessed using a detailed knowledge of the specimen temperature history. In cases where either poor temperature control, often within the + 0.5~'C band, and poor initial bending have been identified then the result has been rejected and the test repeated. In a blind repeatability test carried out on a specimen with 50 mm gauge length and slitted extensometer ridges, variation in measured properties were as follows: lifetime 6.0%: creep strain at failure 8.3%; and secondary creep rate 9.5%. Hence, the experimental procedures employed to achieve repeatability in measured material properties have been shown to be satisfactory. 2.4. Unslitted ridged testpiece The dimensions of the testpieces tested are shown in Fig. la. Four stress levels were selected to give a range of lifetimes for which the test programme could be completed within a convenient but realistic timescale; the stress levels selected were: 275.0, 262.0, 250.0 and 241.3 MPa. The measured experimental creep curves obtained from the tests carried out on testpieces with a gauge length of 50 m m are shown by the continuous lines in Fig. 3. This particular gauge length has been selected since it imroduces the lowest errors in the measured creep strains without recourse to using an unusually long testpiece. These tests were carried out in order to determine the constants in the true constitutive equations. To investigate the effect of gauge length on the measured creep curves, three tests were carried out with gauge lengths of 50, 30 and l0 mm, all at the same stress level of 250.0 MPa. The measured creep curves are shown in Fig. 4. It may be seen that the creep curve for thc testpiece gauge length of 50 mm has creep rates which are typically twice those for the creep curve measured for a gauge length of 10 mm. Also, the lifetime for the testpiece with a gauge length of 50 mm is 76 h, and that for the gauge length of 10 mm is 187 h. Tbi,, r~:~!~ t~,,,~

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the dramatic strengthening effect provided by the testpiece with the shorter gauge length. The strengthening effect is due to the elevation of the first stress invariant { = a,} and the resultant suppression of the effective stress ae { = (3SoSo/2) ~, where g~ is the deviatoric stress tensor} in the region of the extensometer ridges. When the extensometer ridges are sufficiently close, an interaction takes place between the perturbed fields at each ridge. The experimental curve for the 30 mm gauge length is of very little difference to that for the 50 mm gauge length; this is due to the extensometer ridges being sufficiently well separated in both cases to avoid the interaction between the stress and strain fields generated by the extensometer ridges. 2.5. Slitted ridged testpieces To investigate the effect of gauge length on the creep curves measured by the slitted testpieces, tests have been carried out at a stress level of 250.0 MPa for the three gauge lengths 50, 30, 10 mm used with the unslitted testpieces. The resulting creep curves are shown in Fig. 5 by the broken lines. It may be seen that the measured creep rates in all three tests are much closer than those for the unslitted testpieces given in Fig. 4. The lifetimes are much closer than those for the unslitted testpieccs; e.g. in the case of the 50 mm gauge length

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the lifetime is 100 h and that for the 10 mm gauge length is 117 h for the slitted ridges, compared with 76 h and 187 h, respectively, for the unslitted ridges. This clearly shows that the effect of the slitted extensometer ridges is to reduce the degree of interaction between the stress and strain fields at the ridges, particularly in the case of the 10 mm gauge length. Again, for the 50 and 30 mm gauge length testpieces the creep curves are quite close to each other for the same reasons discussed in the case of the unslitted testpieces. The solid lines of Fig. 5 denote the creep test results obtained for the unslitted testpieces with gauge lengths 50, 30 and 10 mm, which have previously been presented in Fig. 4. In connection with the creep strain rates measured for slitted and unslitted'ridged testpieces it may be seen that for the unslitted ridged testpieces the creep rates are lower by a factor of ca 50%. This is due to the reduction in the reinforcement effect of the ridges and its alleviation in the case of the slitted extensometer ridge testpieces. In connection with the measured lifetimes, it may be seen that for the slitted ridged testpieces the range of lifetimes is dramatically reduced producing a spread for the slitted ridged testpieces of 100-117 h compared with a spread of 76-187 h for the unslitted ridged testpieces. This clearly demonstrates the improved accuracy of strain and lifetime measurements made by the slitted ridged testpiece. 2.6. Examination of failed specimens Figure 6 shows failed testpieces for all three gauge lengths tested, with the unslitted ridged testpieces being denoted by the capital letters without the superscript prime, and the slitted ridged testpieces being denoted by the capital letters with the superscipt prime. In all cases it may be observed that all specimens failed without significant necking and that the final failure plane is initiated at a distance from the extensometer ridges equal to ca 1.5 × the diameter of the parallel sided section of the testpiece, except for the testpieces with 10 mm gauge length. This observation is consistent with the theoretical predictions of Dyson et al. [8]. This failure initiation point is associated with the boundary between the zone of perturbation created by the extensometer ridge and the uniform field of the parallel sided section of the testpiece. The short gauge length unslitted testpiece (10 ram) denoted by A has a 45 ° localized shear band failure mechanism which is different from that shown by all the other testpieces, and in particular that for the slitted ridged testpiece of the same gauge length. This is clearly due to the perturbations in the stress, strain and damage fields provided by the unslitted ridges. 2.7. Influence of necking on measured creep curve For those classes of materials in which necking occurs, the interpretation of the creep curve over the period when necking is pronounced is complicated by the nature of the

Uni-axial creep testpieee

761

tri-axial stress-state generated at the notch. This usually does not present severe difficulties since the degree of necking required to produce significant tri-axial stress is large, and this does not take place until the final stages of the test, and can therefore often be neglected. In all specimens tested in this investigation no necking took place; and hence these complications were not encountered. 3. THEORETICAL METHOD 3.1. Constitutive equations

To carry out theoretical computations to predict the behaviour of the slitted and unslitted testpieces it is necessary to know the constitutive equations for the aluminium alloy. The relevant equations at the test temperature have been discussed by Kowalewski et al. [4] and are summarized here. The form of the constitutive equations for multi-axial conditions are given by the following equation set: deu 3 A d--t- = 2 (1 -- co2)~

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(i)

dig) Kc ~ - = T (1 - q,)" dco2 DA f t T t ~ N . . fB(7,(l --_U)~ ~- =(I--~2)"~,~-~) s,nn~, i-'(1) }'

where A, B, H*, h, Kc, D and v are material constants, and n is given by the expression:

which approximates to (Bge(1 - H))/(1 - O) for most cases of interest. Where gu and (re are, respectively, the previously defined deviatoric and effective stresses, and al is the maximum principal stress. The parameter N in the last equation is used to indicate the state of loading; e.g. for al tensile, N = 1; and for ol compressive, N = 0. Material parameters which appear in this model may be divided into three groups, i.e. (i) the constants h and H* which describe primary creep; (ii) the parameters A and B which characterise secondary creep; and (iii) the parameters K, and D responsible for damage evolution and failure. The second equation in set (1) describes primary creep using variable H, which varies from 0 at the beginning of the creep process to H*, where H* is the saturation value of H at the end of primary period and subsequently maintains this value until failure. The equation set contains two damage state variables used to model tertiary softening mechanisms. The first damage state variable, O, which is described by the third equation in set (1), is defined from the physics of ageing to lie within the range 0-1 for mathematical convenience. The second damage variable, co2, defined by the fourth equation in set (1), describes grain boundary creep constrained cavitation, the magnitude of which is strongly sensitive to alloy composition and to processing route. Creep constrained cavitation can either be nucleation or growth controlled. Irr this paper, only nucleation control will be considered. Using a nucleation rate model which is linear in strain rate; proportional to the stress parameter (al/tre)v; and a failure criterion of co2 = 1/3 gives: do~2 = (trl") v 8e

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(2)

where e2 = 2eueu/3, eI is the strain at failure under uni-axial tension, and is a material constant which describes the stress-state sensitivity of grain boundary cavity nucleation. The effect of stress state on cavity nucleation, described by (6l/ae) ~, is based on the work of Dyson and McLean [9].

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3.2. Normalisation of constitutive equations In order to use Eqns (1) in Finite Element Continuum" Damage Mechanics studies, and to maintain accuracy and stability of numerical solutions, it is appropriate to normalisc the equations by introduction of the following terms: ~-~ij = (7ij"(70~

/'ij = Eij./Ut)"

Sij = Sii,,;ao .

where eo = cro/E and E is the elastic modulus. The normalised time may bc defined as: r =

(EA.!tro)dt = ( E A i a o ) t = (A/eo)t.

Equations (1) may be rewritten using these parameters as: dJij --= dr

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The equations will be used together with the effective stress multi-axial rupture criterion Ibr the aluminium alloy, given by t = 0, as discussed by Kowalewski et al. [4]. The equations have been used to determine the time variation of the stress, strain and damage fields within the testpieces, and the techniques used are discussed in the following section. 3.3. Continuum damage mechanics finite element soh,er Creep damage describes the material degradation which gives rise to the acceleration of creep rate known as tertiary creep. The material is treated as a continuum. In a continuous body under load, the constitutive equations are used to describe how the stress, strain rate and damage are interrelated at a point (or at an element) in the material. Different points in the body suffer different stress levels and are at different stages of material deterioration. The constitutive equations remain valid until the specified rupture criterion is reached. At this time, the point under consideration can no longer sustain load and, at this point Ior element), the material has failed. Ultimately, a region of failed material will progress through the body until the applied load can no longer be sustained and the body fails as a whole. The numerical procedure used to solve the boundary value problems for creep-damage deformation is that used by Hayhurst et al. [6,1. It is based on the finite element method and employs constant-strain triangular elements which have been extensively used to model the behaviour of notched and cracked creep components [10-1. The finite element mesh generator package, F E M G E N , is used to generate the meshes. The solution method takes the elastic solution as its starting point and integrates the creep strains, 2 u, and creep damage-state variables 6,91,to2, with respect to normalised time. The integration is carried out over a series of discrete normalised time steps using a fourth-order Runge Kutta technique; this procedure involves the repeated solution of the boundary valuc problem to determine the field quantities required for the numerical solution. Creep damage, as represented by the two damage-state variables, develops monotonically with time throughout the structure, and failure of an element is deemed to have occurred when the second

Uni-axial creep testpiece

763

damage state variable, co2, attains the prescribed value. The material element is then unable to transmit or sustain load and it is removed from the model. The boundary value problem is then redefined to allow either a crack, or damage zone, to develop and spread. Once the boundary value problem is redefined, the time integration is continued by taking the new starting point as the field variables before the local failure occurred. The procedure is then repeated until complete failure of the cracked member occurs. 3.4. Numerical techniques The geometry of the uniaxial tensile creep testpieces are shown in Fig. la with unslitted extensometer ridges and in Fig. lb with slit extensometer ridges. Each of the slitted extensometer ridges has 24 slits to form 24 ridge portions, which are used to release the constraints in the hoop direction on the ridges. The number of slits was set at 24 each to ensure a sufficiently small circumferential slit width to achieve plane-stress conditions circumferentially. A single quadrant of the testpiece [2] containing the extensometer ridge was selected for investigation and divided into two regions: region one for the main part, including the parallel section, of the testpiece, and region two for the extensometer ridge area. The quadrant of the testpiece was subdivided into axisymmetric triangular elements, and their size was redefined in the zone around the testpiece extensometer ridges where a high stress concentration occurs. A coarse mesh was selected in the parallel portion of the testpiece where the stresses are uniform and do not change much. The mesh which was generated for the quadrant of the testpiece contained 566 constant-strain triangular elements, including 60 elements in region two and 506 elements in region one, respectively, and 322 nodes. The elements were subjected to axisymmetric stressing in region one, and to plane stress in region two due to the release of hoop constraints by slitting. The D matrix, which is used to express the stress-strain relationship for the axisymmetric loading situation, was redefined I-2] by use of the plane-stress condition E:z = 0 where Z is the hoop direction, to a D, matrix for the elements in region two. Both D and D, are 4 × 4 matrices in axisymmetric form. The linear multiplier a( = 10 -6) was introduced in the D, matrix to ensure that the hoop stress Ezz was small. In addition to using the D, matrix, Young's modulus E for region two was divided by 50. This figure was calibrated by comparing experimental results with theoretical predictions for the 10 mm gauge length slitted ridged testpiece. The half-band width of the stiffness matrix was optimised by automatically renumbering the element nodes. This was done to achieve efficient usage of computer resources, and to enhance the accuracy of the numerical computations. The optimised mesh was then input to the Continuum Damage Mechanics solver DAMAGE XX. The load boundary of the mesh was subjected to a constant normalised stress to ensure that the normalised stress level at the gauge area Err was equal to unity, and the boundaries of symmetry were subjected to zero normal displacement with Exr = 0. The computations were carried out on an IBM 3090-600E computer at the Science and Engineering Research Council, Rutherford Appleton Laboratory. The computed results of the stresses, strains, displacements and damage-state variables were stored for subsequent examination by the postprocessor FEMVIEW. The computer modelling results of the elastic stress redistribution and the accuracy of the measured strain taken from the testpiece with slit extensometer ridges are presented in the next section. 4. DETERMINATION OF TRUE CONSTITUTIVE PARAMETERS The results of experimental tests on unslitted ridged testpieces with 50 mm gauge length carried out with stress levels of 241.3, 250.0, 262.0, 262.0, and 275.0 MPa, (Fig. 3) have been used to determine the material constants in the constitutive equations, Eqns (1), using the optimisation scheme reported by Kowalewski et al. [4]. The constitutive parameters determined were employed in the Continuum Damage Mechanics solver DAMAGE XX to carry out finite element computations for the unslitted ridged testpieces for the four stress levels given. The true axial component of strain, e , , in the testpiece gauge length at any time

Z.L. KOWALEWSK!et al.

764

t, was determined as the volume average given by: C.y>. V /

gvv = i=1

i=1

where i is the finite element number and m is the number of elements falling along the line joining the surface of the specimen with the centreline in the plane of testpiece symmetry; the superscript e denotes finite elemental values, and V denotes the volume of a finite element. This mean axial strain is considered as the uniformly distributed creep strain in the parallel sided region of the testpiece. The axial creep strain measured by the extensometers located on the testpiece ridges is denoted by the symbol grr, determined from,

~,'r=

6 i Y,

where 6 is the axial displacement of the extensometer ridges (Fig. la) and Y is the guage length defined by the axial separation of the extensometer ridges. The accuracy of the creep strain measured using the extensometers may be assessed using the R-ratio value [1 ], which is defined by: R = %r/gr~,.

The variation of the R-ratio with the normalised time fraction t / t f is given in Fig. 7 by the solid line for the unslitted ridged extensometer testpieces of gauge length 50 mm at the stress 250.0 MPa. The time to failure is denoted by t f . Examination of the R-curves for other stress levels with the same gauge length are not considered here since the curves are independent of stress. The shape of the R-curve is similar to that of a classical creep curve, and is totally different from the variation determined for a Nimonic superalloy by Lin et al. [1], where a uni-modal peaked distribution was obtained. The reason for this is the location of the mode of failure. In the case of the aluminium testpiece studied here failure takes place in the centre of the testpiece, whereas in the case of the Nimonic superalloy testpiece failure takes place in the region of the extensometer ridge. The same shape of R - t / t f curve has been obtained for large deformation, high stress studies of Nimonic superalloys [8] where the failure mode takes place in the parallel sided region of the testpiece. The continuous R - t / t f curve given in Fig. 7 has then been used to correct the experimental creep curves presented in Fig. 3, with the objective of generating a set of creep curves which are closer to the true curves for the aluminium alloy. The new set of creep

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curves which resulted from this process were then used as input to the optimisation scheme reported by Kowalewski et al. 1"4] to redetermine the constitutive parameters. The constitutive parameters so determined were then used in the solver D A M A G E XX to generate the second R - t / t I curve shown by the broken line in Fig. 7. This second curve was then used to recorrect the original experimental creep curve shown in Fig. 3, and the process repeated to produce the third R - t/t I curve which is shown by the chain line in Fig. 7. It may be observed that the second and third R - tit s curves are almost identical except for small errors in the latter stages, 0.4 ~< t i t I ~< 1.0, of the curve. If a further correction was to be carried out, then the corrections to be made to the original creep curves of Fig. 3 would be negligible. The process may therefore be considered to have converged and the true constitutive parameters obtained, for which the constants are given in Table 1. The final modifications to the experimental creep curves given in Fig. 3 are shown in Fig. 8 by the continuous lines, and the broken lines denote the predictions made by the fitted data listed in Table 1. The true constitutive equations, given by Eqns (1) and the data of Table I, will be used in the following subsections to predict the behaviour of the unslitted and slitted ridged testpieces. 5. C O M P U T A T I O N A L

RESULTS

5.1. Creep curves f o r unslitted ridged specimens

The true constitutive equations were used as input to the Finite Element Continuum Damage Mechanics solver D A M A G E XX to predict the behaviour of unslitted creep test testpieces for three different gauge lengths: 50, 30 and 10 m m subjected to the same stress level of 250.0 MPa. The theoretical creep curves, computed from the extensometer displacements, for the unslitted ridged testpieces are presented in Fig. 9 where they are denoted by the broken lines, and compared with the experimental results, denoted by the continuous lines, previously presented in Fig. 4. The predicted creep curves for the 50 and 30 m m gauge

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lengths are almost identical, but the creep curve predicted for the 10 mm gauge length specimen has creep rates which are much lower, and lifetimes which are much longer than those for the 50 and 30 mm gauge lengths. The theoretical and experimental curves for the 10 mm gauge length are in close agreement, whereas the experimental curves for the 50 and 30 mm gauge lengths are in slight disagreement with the theoretical predictions; but, the differences are small, and they are certainly within experimental error. 5.2. Creep curees jbr slitted rid#ed specimens The true constitutive equations have been used in the solver D A M A G E XX to predict the creep curves at the stress level of 250.0 M P a for the slitted ridged testpiece; the curves are shown in Fig. 10 using the broken lines for the three gauge lengths: 50, 30 and 10 mm, where they are compared with the results of experiments denoted by the continuous lines, also shown by the broken lines of Fig. 5. The predicted creep curves for the 50 and 30 mm gauge lengths are very close. The predicted creep curve for the 1 0 m m gauge length specimen, as in the case of the unslitted ridged specimen, shows creep rates which are lower,

Uni-axial creep testpicce

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and lifetimes which are longer than for the 50 and 30 mm gauge lengths. However, the predicted degree of strengthening, expressed as a ratio of lifetime of the 10 mm gauge length testpiece to that for the 50 mm gauge length testpiece is 1.37; this compares with a value of the same ratio of 2.06 for the unslitted ridged testpiece. This theoretical result clearly confirms the experimental observations, and that the presence of the slits in the extensometer ridges relieves the constraint to deformation provided by the unslitted extensometer ridges. By comparison of the predicted creep curves (broken lines) of Figs 9 and 10, in the primary and secondary stages, it may be seen that the creep strains for the 10 mm gauge length slitted ridged testpiece have increased relative to those for the unslitted ridged testpiece, hence producing results closer to the expected curves, approximated to those given for the 50 and 30 mm gauge length testpieces. Comparison of strain levels in Fig. 10 for the theoretical curves with those for the experimental curves illustrates how primary and secondary creep are closely predicted; but, how the tertiary creep behaviour is somewhat in error for both the 50 and 30 mm gauge lengths. The errors between the theoretical and experimental curves for the 10 mm gauge length are small due to the calibration of the finite element properties for the extensometer ridge elements discussed in Section 3.0. This clearly shows that, within experimental error, the true constitutive equations may be used to predict the creep strain behaviour of the slitted ridged testpieces. 5.3. Distribution of field variables within the testpiece (a) Creep damage co2. Figure 11 shows the distributions of the creep constrained cavitation damage parameter co2 at the time fraction t / t / = 0.999 for the unslitted and slitted ridged testpieces, denoted by the lower case identifiers -us and -s, respectively, with gauge lengths of 50 and l0 mm. It can be seen that the failure of the unslitted and slitted testpieces, indicated by the zones of high damage, takes place in the parallel gauge sections for both 50 and 10 mm gauge lengths. By comparison of the failure modes of all testpieces, it may be observed that the same failure modes occur for the 50 mm gauge length slitted and unslitted testpieces, and for the l0 mm gauge length slitted testpiece. A different failure mode has been predicted for the 10 mm gauge length unslitted ridged testpiece, which has a highly localised damage zone emanating from the root radius of the extensometer ridge; this result is supported by the experimentally observed failure mode shown in Fig. 6, part A, which indicates a 45 ° localised shear band failure mechanism. This is due to the higher local stresses, and to the strengthening effect provided by the close proximity of the extensometer ridges in the shorter gauge length testpiece. Comparison of the regions of high damage in Fig. I l with testpiece failures shown in Fig. 6 for the 50 and 30 mm gauge lengths shows that failure is not associated with the extensometer ridges; and that the predicted failure locations do not coincide with those observed experimentally. Failure does not take place in the experiments at the centre of the gauge length as predicted theoretically. The inability of the theory to predict the precise location of failure within the gauge section is consistent with the work of Dyson et al. [8] who showed that it is necessary to carry out large strain analyses to accurately predict failure location. However, this deficiency of the theoretical predictions relates only to final failure; the accuracy of the predicted strain-time histories is not influenced. (b) Normalised effective creep strain 2e. The distributions of the normalised effective creep strains 2e { = ee/eo } at the time fraction t/t: = 0.7 for the unslitted, -us, and slitted, -s, ridged testpieces are shown in Fig. 12 for gauge lengths of 50 mm and 10 mm. The strain distributions for the 50 mm gauge length testpieces are uniform along the majority of the gauge length for both unslitted and slitted testpieces; but, this is not entirely the case for the unslitted and slitted testpieces with 10 mm gauge lengths. For the 10 mm gauge length testpiece the volume of material subjected to high levels of "uniform" strain denoted by the dark red regions is increased by approximately three times in the slitted testpiece compared with the unslitted testpiece. Despite the lack of perfect uniformity in the 2, field, this increase is the main reason for the dramatic improvement in accuracy. In the field plot of ;re for the HS 36:8-F

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unslitted testpiece, the first indication of localised strain may be observed at the toe of the extensometer ridge root radius. This is probably linked to the 4Y" shear failure mechanism discussed in the previous section and in Section 2.3. However, for both gauge lengths, the effect of slitting the extensometer ridges can be seen to substantially increase the extent of the uniformity of the strain field. 5.4. The R-curve: errors in strain m e a s u r e m e n t The R - t / t y curves computed with the true constitutive equations are given in Fig. 13 for the unslitted ridged testpieces, continuous curves, and for the slitted ridged testpieces. broken curves, with the gauge lengths of 50, 30 and 10 mm; the curve for the 50 mm gauge length, with unslitted ridges, corresponds with the chain line of Fig. 7. The vertical downward shift of the unbroken curves to the broken curves indicates the improved accuracy achieved. This effect is most pronounced for the 10 mm gaugc length. These observations confirm those made from Figs 9 and 10. However, despite this downward shift the remaining errors for the slitted ridged specimens are significant, a typical average valuc of R for the unslitted ridged 10 mm gauge length testpiece being R = 1.30. This indicates that further improvements are required in the testpiece design. 6. D I S C U S S I O N A N D C O N ( ' L U t S I O N S

Comparison of Figs 9 and 10 for the 50 and 30 mm gauge lengths with unslitted and slitted extensometer ridges shows that both theoretical and experimental curves are in reasonably good agreement. This shows that the errors introduced by the extensometer ridges are small compared with the measured strain levels. Comparison of the creep curves in Figs 9 and 10 for the 10 mm gauge length show an improvement in Fig. 10 which is entirely due to the effect of reducing the circumferential compression stresses in the ridges by introduction of the slits. However, this effect is not sufficiently strong enough to yield a creep curve for the 10 mm gauge length which is close to that for both the 50 and 30 mm gauge length testpieces, and further work is required to redesign the slitted testpiece. The effect of the slitted extensometer ridges on the accuracy of the measured creep strains, on the failure mode, and on the lifetimes becomes more pronounced for the shorter testpiece gauge lengths (Figs 9-11). It is recommended that the testpiece with slitted extensometer ridges should be selected in combined cyclic plasticity and creep testing, where a shorter gauge length has to be used. Clearly the issue of whether the presence of the slitted ridges will initiate low cycle fatigue failure will have to be addressed, and may provide limitations particularly for low temperature behaviour.

Uni-axial creep testpiece

769

T h e n u m e r i c a l p r o c e d u r e s d e v e l o p e d b y L i n et al. [ 2 ] for m o d e l l i n g t h e c r e e p b e h a v i o u r of the testpiece with slitted extensometer ridges have been validated by comparison of the n u m e r i c a l p r e d i c t i o n s w i t h t h e e x p e r i m e n t a l results. C l o s e a g r e e m e n t h a s b e e n o b t a i n e d as s h o w n i n F i g s 9 a n d 10. Acknowledgements--The authors gratefully acknowledge the support of the British Council and UMIST for funding the research visit of Z. L. Kowalewski. Also one of the authors, D. R. H. acknowledges support of the SERC of Great Britain for the provision of computer resources. The ongoing discussions with Dr B. F. Dyson, Division of Materials Metrology, NPL, which took place throughout the course of this research are gratefully acknowledged. The help and support of Mr C. J. Morrison with materials and creep data has been invaluable to the progress of the research. Without the skills and participation of Mr J. E. Boon, Mr P. Baldwin and Mr K. Dabbs the specimens used in this research could not have been produced; their work is gratefully acknowledged.

REFERENCES 1. J. LIN, D. R. HAYHURSTand B. F. DYSON,The standard ridged uniaxial testpiece: computed accuracy of creep strain. J. Strain Analysts 28, 101 (1993). 2. J. LIN, D. R. HAYHURSTand B. F. DYSON,A new design of uniaxial creep testpiece with slit extensometer ridges for improved accuracy of strain measurement. J. Mech. Sci. 35, 63 (1993). 3. Y. OHASH1,M. TOKUDA and H. YAMASHITA,Effect of third invariant of stress deviator on plastic deformation of mild steel. J. Mech. Phys. Solids 23, 23 (1975). 4. Z. L. KOWALEWSKI,D. R. HAYHURSTand B. F. DYSON,Mechanisms-based creep constitutive equations for an aluminium alloy. J. Strain Analysis (in press) (1994). 5. F. R. HALL and D. R. HAYHURST,Continuum damage mechanics modelling of high temperature deformation and failure in a pipe weldment. Prec. R. Soc. Lend. A. 433, 383 (1991). 6. D. R. HAYHURST, P. R. BROWN and C. J. MORRISON,The role of continuum damage in creep crack growth. Phil. Trans. R. Soc. Lend., A 311, 131 (1984). 7. D. R. HAYHURST,The effect of test variables on scatter in high temperature tensile creep-rupture data. Int. d. Mech. Sci. 16, 829 (1974). 8. B. F. DYSON, D. R. HAYiiURSTand J. LIN, The role of large deformation on the mode of failure and on the accuracy of measured creep strains of a ridged uniaxial testpiece (submitted) (1994). 9. B. F. DYSON and M. MCLEAN,Creep deformation of engineering alloys: development from physical modelling. ISIJ Int. 30, 802 (1990). 10. D. R. HAYHURST,P. R. DIMMER and C. J. MORRISON,Development of continuum damage in creep rupture of notched bars. Phil. Trans. R. Soc. Lend. A311, 103 (1984).