- Email: [email protected]
Unusual nonlinear response of some metallic materials
Mechanics of Materials 31 (1999) 565±577
www.elsevier.com/locate/mechmat
Unusual nonlinear response of some metallic materials Christopher M. Gil a, Cli J. Lissenden b
a,*
, Bradley A. Lerch
b
a Department of Engineering Science and Mechanics, Pennsylvania State University, University Park, PA 16802, USA Life Prediction Branch, Lewis Research Center, National Aeronautics and Space Administration, Cleveland, OH 44135, USA
Received 28 July 1998; received in revised form 14 April 1999
Abstract An anomalous material response has been observed under the action of compressive loads in ®brous silicon carbide/ titanium (both Ti-6242 and Ti-15-3 alloys) and the nickel-base alloy Inconel 718 in the aged (precipitation hardened) condition. The observed behavior is an increase, rather than a decrease, in the instantaneous Young's modulus with increasing load and is referred to as stiening. This increase is small, but can be signi®cant in yield surface tests where an equivalent oset strain on the order of 10 le (10 ´ 10ÿ6 m/m) is being used. Stiening has been quanti®ed by calculating oset strains from the linear elastic loading line. The oset strains associated with stiening during compressive loading are positive and of the same order as the target oset strains in yield surface tests. Stiening appears to be nonlinear elastic and for SiC/Ti-15-3 decreases as the test temperature increases, suggesting a temperature-dependent mechanism. No stiening occurred in SiC/Ti-15-3 above approximately 500°C. In addition, the tensile response of aged Inconel 718 exhibits a nonlinear elastic response (not stiening) prior to the initiation of plastic deformation. The nonlinear elastic response in both tension and compression agrees with a second order stress±strain relation. The higher order terms may be important due to constraint interactions associated with a strengthening second phase such as precipitates or ®bers. Ó 1999 Elsevier Science Ltd. All rights reserved. Keywords: Yield surface; Plastic ¯ow; Tension test; Compression test; Stiening; Nonlinear elasticity; Metal matrix composite; Precipitation hardened alloy
1. Introduction The experimental work described herein stemmed from an ongoing eort to map out ¯ow surfaces for metallic materials in the axial±shear stress plane at elevated temperatures. A ¯ow surface is similar to a yield surface, except it applies to ratedependent materials. This eort is in support of the development and validation of constitutive models for advanced, structural materials in
* Corresponding author. Tel.: 804-863-5754; fax: 804-8636031; e-mail: [email protected]
aeronautics applications. These experiments involve loading a specimen biaxially up to the onset of yielding and then unloading before signi®cant plastic ¯ow occurs. This is repeated several times on the same specimen using dierent stress ratios until the yield surface has been mapped out, after which the data can be analyzed to obtain ¯ow surfaces. The de®nition of yielding that is used in these experiments in¯uences the resulting yield surface (Khan and Wang, 1993; Hecker, 1976). The de®nition of yielding chosen for this study involves detecting oset strains `on the ¯y'. That is, osets from the linear elastic loading line are determined
0167-6636/99/$ ± see front matter Ó 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 6 3 6 ( 9 9 ) 0 0 0 1 8 - 6
566
C.M. Gil et al. / Mechanics of Materials 31 (1999) 565±577
under load, not after unloading (permanent set). This method greatly expedites detection of yielding, but relies on the assumption that the oset strains are directly related to permanent set. A one-to-one relationship between the maximum oset strain and the permanent set is expected if the initial loading and unloading moduli are equal. By seeking a very small oset strain, on the order of 10 ´ 10ÿ6 m/m (or 10 le), the change in the internal state is minimized and multiple loading excursions (probes) can be conducted on the same specimen to map out the yield surface. Additionally, it removes any concerns about specimen to specimen scatter in the yield surface since the same specimen is used for each probe. It is also cost eective in terms of the number of specimens required. Similar multiprobe single-specimen yield surface experiments have been successfully conducted by many researchers on solution annealed monolithic materials (e.g., Phillips et al., 1972; Liu and Greenstreet, 1976; Helling et al., 1986; Wu and Yeh, 1991) as well as on an annealed boron-aluminum composite (Dvorak et al., 1988; Nigam et al., 1994a, b). Furthermore, the experimental setup used in the current study was implemented by Lissenden et al. (1997) to determine ¯ow surfaces for solution annealed type 316 stainless steel at elevated temperature. However, attempts to determine ¯ow surfaces on a unidirectional SiC/Ti6242 tube have been unsuccessful to date due to an anomalous response, termed stiening, under compressive loading. Stiening is characterized by an increase in the stiness during loading, which creates an oset from the initial loading curve that is often comparable in magnitude, but opposite in sign, to the target value of the strain oset that is sought (Fig. 1). The observation of stiening led to a re-evaluation of the strain measurement apparatus and experimental procedure. After a thorough investigation, the equipment was determined to be functioning properly. In the process of validating the experimental setup and procedures, tubular Inconel 718 specimens were tested. It was found that precipitation hardened (aged) Inconel 718 (IN-718A) exhibited stiening, while solution annealed Inconel 718 (IN-718S) did not.
Fig. 1. Schematic of the Stiening phenomenon in a stress± strain response.
To further study stiening under pure axial loading, thick SiC/Ti-15-3 plates and IN-718A bars were cut into dogbone specimens. Various loading sequences were applied to these specimens. All of the tests described in this paper were aimed at explaining the cause of stiening observed during a yield surface probe.
2. Experimental procedure Quasi-static loading was applied to the various specimens as described below. 2.1. Test equipment Experiments were conducted on one of two biaxial servohydraulic test machines having an axial load capacity of 222,500 N (50,000 lbs) and a torque capacity of 2260 N m (20,000 in lbs). Rotational and translational movements of the actuator are independently controlled by a personal computer coupled with the electronic controller on the machine. The controller allows for rotational control by angle, torque, or shear strain and translational control by displacement, load, or axial strain.
C.M. Gil et al. / Mechanics of Materials 31 (1999) 565±577
Axial and shear strains were measured by an extensometer and in some cases strain gages were also used. The extensometer is a standard, o-theshelf, water-cooled biaxial model with a gage length of 25 mm. Precision strain gages (MicroMeasurements Type EA-06-125BZ-350) having an electrical resistance of 350 X and a gage length of 3.2 mm were installed on select dogbone specimens that were to be tested at room temperature. Four strain gage rosettes were mounted on the type 316 stainless steel tubular specimens (Lissenden et al., 1997). A strain gage conditioner (excitation voltage of 4 V, calibration factor of 1000 le/V) capable of handling multiple strain gages was used. A 5 kW induction heating system with three adjustable coils, described by Ellis and Bartolotta (1997), was used to heat the specimens. Three type K thermocouples were spot welded to the gage section of the specimen, with the center thermocouple used to control the temperature. The two remaining thermocouples were used to ensure a thermal gradient in the gage section of within 1% of the desired test temperature. 2.2. Specimens 2.2.1. Fibrous silicon carbide/titanium One tubular SiC/Ti-6242 specimen was used in this study. The SCS-6 ®bers are oriented along the axis of the tube, which was fabricated using the foil±®ber±foil technique. The exact ®ber volume fraction has not yet been measured, but is nominally 0.25. The tube has inner and outer diameters of 20.7 and 23.3 mm, respectively, and was cut to a length of 203 mm. It was gripped by a ®xture that was designed to transfer stress as uniformly as possible without crushing or splitting the tube. A total of seven [0]32 SiC/Ti-15-3 dogbone specimens were tested in tension and compression. Each specimen contained 35% by volume SCS-6 silicon carbide ®bers aligned longitudinally to reinforce a Ti-15-3 matrix. The specimens were processed by consolidating alternating layers of Ti-15-3 foil and SiC ®bers to achieve a total thickness of approximately 7.6 mm. This thickness was sucient to obviate buckling under the applied compressive loads. The specimens were aged at 700°C for 24 h in a vacuum. Each specimen was
567
203.2 mm long and had the geometry shown in Fig. 2. At each end, stainless steel tabs were bonded to either side of the specimen to ®ll the collet gap in the grips and ensure that the proper gripping pressure was applied. Approximately 44.5 mm of each end of the specimen was gripped. 2.2.2. Monolithics The solution annealed type 316 stainless steel tubular specimen, described in detail by Lissenden et al. (1997), is 229 mm long and has a reduced gage section. The gage section has inner and outer diameters of 22 and 26 mm, respectively. The Inconel 718 tubular specimens are 229 mm long and have inner and outer diameters in the reduced gage section of 15.9 and 21 mm, respectively. After machining, the Inconel-718 specimens were solutioned at 1038°C in argon for 1 h and air cooled. Select specimens were further heat treated as follows: aged at 720°C in argon for 8 h, cooled at 55°C per h to 620°C and held for 8 h, then air cooled to room temperature. Specimens subjected to only the ®rst of these heat treatments are referred to as solutioned (IN-718S), while specimens subjected to both heat treatments are referred to as aged (IN-718A). Two dogbone specimens were machined out of the same Inconel 718 extruded bar stock from which the tubes were machined. The ®nal specimen design was a ¯at, dogbone specimen having a 6.4 ´ 6.4 mm2 gage cross-section as shown in Fig. 2. These specimens were given the same aging treatment described above for the IN-718A tubes. One longitudinal and one transverse strain gage
Fig. 2. Specimen designs for SiC/Ti-15-3 and IN-718A dogbone specimens.
568
C.M. Gil et al. / Mechanics of Materials 31 (1999) 565±577
were mounted at the center of each of the four sides of these two specimens. 2.3. Test procedure Dogbone specimens. Each test was run in axial load and rotation control with the rotation remaining ®xed throughout the test (i.e., a standard uniaxial test), except axial strain control was used for IN-718A specimens. The axial load (engineering stress) was increased monotonically and the initial Young's modulus, E, was determined by linear regression during a prede®ned stress range. A constant data sampling interval was used throughout the test. The stress range was determined by past experience and represented a balance between gathering enough data to accurately calculate the modulus and yet not exceeding the proportional limit. Once the modulus was determined, the strain oset from the linear elastic loading line, eoff , was calculated from the total strain, e, the applied stress, r, the initial stress, r0 , and the modulus, r ÿ r0 1 eoff e ÿ E and continually compared to a target value (usually 10 le). When the oset strain reached the target value the specimen was unloaded at a rate four times that of the loading rate, except an unloading rate equal to the loading rate was used for IN-718A specimens. The increased unloading rate was used to expedite the tests since several repeats of each test were commonly performed. This type of test is typical of what is done in the mapping of a yield or ¯ow surface (Lissenden et al., 1997). The primary loading sequence was tension until the oset strain reached a target value, followed by compression until the oset strain reached a target value. Results are presented for IN-718A specimens tested at room temperature, in which both strain gage and extensometer data were collected. Tests on SiC/Ti-15-3 at 427°C, 482°C, 538°C and 649°C are the only elevated temperature results that are reported in this paper. No strain gages were used at elevated temperature. Tubular specimens. Proportional axial±torsional loads were applied to the type 316 stainless steel
tube and the SiC/Ti-6242 tube using the procedure described in detail by Lissenden et al. (1997). We present results from the Inconel 718 tubular specimens for uniaxial compression only. For the current report the nature of the biaxial loading is unimportant and will not be discussed further. The axial oset strain was determined using Eq. (1). 3. Results Experimental results for dierent materials will be presented in the order in which the tests were conducted: 1. a type 316 stainless steel tubular specimen, where the material stress±strain response was as expected, 2. a SiC/Ti-6242 tubular specimen, where anomalous material stress±strain response was ®rst detected, 3. SiC/Ti-15-3 dogbone specimens, where the anomalous stress±strain response was studied in detail, and which represent the bulk of the results presented in this report, 4. IN-718S (no stiening) and IN-718A (stiening) tubular specimens, and 5. IN-718A dogbone specimens that exhibited nonlinear elastic behavior in both tension and compression and represent the most informative results. 3.1. Type 316 stainless steel Results from yield surface experiments on solution annealed 316 stainless steel are presented by Lissenden et al. (1997). Here we will only look at the axial stress±strain response from one typical probe of a yield surface at room temperature (having an eective loading rate of 1 MPa/s). The axial responses from other probes, including nearly pure uniaxial compression, were similar. Strains were measured during proportional loading probing experiments with the extensometer and strain gages. The axial stress±strain response from the ®rst probe (a 12° probe angle in the axial± shear stress plane; that is, predominantly axial tension) of surface determination run number six is shown in Fig. 3; shear stresses (s 0.2126r) and
C.M. Gil et al. / Mechanics of Materials 31 (1999) 565±577
569
and strain gage measured strains after the load reversal (Fig. 3) was generally not observed in the oset shear strain data. 3.2. SiC/Ti-6242 tubular specimens
Fig. 3. Solution annealed type 316 stainless steel axial response to a yield surface probe at 12° in the axial±shear stress plane.
strains were also present but are not shown here. The initial response is linear elastic with the linearly regressed stress±strain equations in the strain range 0 < e < 150 le being r 196; 510e 0:11652 extensometer r 190; 970e 0:13285 strain gages;
2
with r in MPa and e in m/m. These equations represent the elastic loading lines from the extensometer and strain gage data, respectively. The stress±strain curve and linear ®t in Fig. 3 are based on the extensometer strain measurement. Fig. 3 also shows the (axial) oset strain as a function of (axial) total strain for both the extensometer and strain gages. The oset strain starts to increase at a total strain of 400 le and once the load is removed the oset strain remains as permanent set. This is typical of plastic deformation in metals and strongly suggests that the oset strain is an excellent indicator of the permanent set. Notice that while the equations of the elastic loading lines were dierent for the extensometer and the strain gages (that is, a 2.8% dierence in Young's modulus), the oset strains measured were nearly identical. Although not shown, for each yield surface probe, each oset strain component had the same sign as its total strain counterpart (excluding scatter). We note that the good agreement between extensometer
Having good success with the type 316 SS specimens, we attempted to conduct yield surface tests on SiC/Ti-6242 tubular specimens at room temperature. Fig. 4 shows the axial stress and the axial oset strain as a function of the total axial strain for a probe angle of ÿ168° (predominantly axial compression, s ÿ0.2126r) and an equivalent loading rate of 2.1 MPa/s. Again, shear stresses and strains were present, but are not shown. Since we had anticipated the center of the ¯ow surface to be shifted in the compressive stress direction due to residual stresses, each probe was started at a compressive stress of approximately 420 MPa. The elastic loading line in the strain range ÿ2500 < e < ÿ2200 le is r 189; 612 e 1:2907 MPa
3
and although the stress±strain response appears to be linear throughout, small oset strains are present. Interestingly, these oset strains have positive, rather than negative, magnitudes and therefore indicate stiening. Additionally, the oset strain decreased during unloading more than it
Fig. 4. SiC/Ti-6242 axial response to a yield surface probe at ÿ168° in the axial±shear stress plane.
570
C.M. Gil et al. / Mechanics of Materials 31 (1999) 565±577
had increased during loading, resulting in a very small (3 le) compressive permanent set. While the stiening observed during loading is very representative of what we observed for compressive loading, and even occasionally for torsional loading, the calculated oset strains during unloading were somewhat erratic. Even though these oset strains are small, they are statistically meaningful. We have determined the 95% con®dence limits on the linearly regressed loading line and extrapolated them up to the maximum load. Doing so, we found that the measured stress±strain response falls above the upper limit. There were 250 data points used in the regression and the correlation factor was 0.99958. In addition, this strain range was very successful for determining the elastic loading line during tensile loading. 3.3. SiC/Ti-15-3 dogbone specimens SiC/Ti-15-3 dogbone specimens, rather than the few available SiC/Ti-6242 tubular specimens, were used to closely examine uniaxial response. These specimens have the same SCS-6 ®ber and a weak interface similar to the SiC/Ti-6242 specimens. Hence, the behavior of the dogbone specimens should be similar to that of the tubes. In addition, the SiC/Ti-15-3 specimens were plentiful and less expensive than the tubes. Unfortunately, these specimens were not entirely ¯at, which complicates the interpretation of the results. 3.3.1. Room temperature Tensile and compressive loadings were applied at room temperature at a rate of 1.67 MPa/s. Unloading was triggered by the oset strain attaining a target value of 10 le in this and all subsequent tests, unless otherwise noted. Additionally, reported strains were measured by the extensometer unless stated to the contrary. The coecients of the elastic loading line, r Ee r0 ;
Table 1 SiC/Ti-15-3 elastic loading line coecients for tensile and compressive loading Test 1 2 3 4
Tension
Compression
E (MPa)
r0 (MPa)
E (MPa)
r0 (MPa)
190,565 189,369 189,151 189,092
0.104337 0.047183 0.721945 0.439698
191,351 190,984 191,216 190,865
ÿ1.78900 ÿ3.85694 ÿ3.37030 ÿ4.08399
the compressive Young's modulus is approximately 1% larger than the tensile Young's modulus in these four tests. The larger (negative) initial stresses for compressive loading are an indication of the small permanent set that occurred due to the tensile loading. The typical calculated oset strain is shown in Fig. 5 as a function of the total strain for tensile and compressive loadings. The oset strain is essentially zero during elastic loading (e< |1000| le) and is positive thereafter. The tensile (compressive) stress present at the load reversal was 451 MPa (ÿ601 MPa). The permanent set turned out to be more than twice the maximum oset during loading. This dierence could be due to measurement problems with the extensometer upon reversal of the load (such as reseating of the extensometer rods in their indents) or due to ratedependence of Young's modulus (a viscoelastic eect) since unloading was done at four times the loading rate. We have no data indicating that Young's modulus is loading rate-dependent at room temperature, but it does appear to be ratedependent at elevated temperature. However, even a very slight rate-dependence can have an eect. If
4
are shown in Table 1 for four consecutive tests on the same specimen. There is a 1% variation in the tensile Young's modulus and a 0.2% variation in the compressive Young's modulus. Additionally,
Fig. 5. Oset strains from both tensile and compressive loadings of SiC/Ti-15-3.
C.M. Gil et al. / Mechanics of Materials 31 (1999) 565±577
the faster unloading rate were to increase the Young's modulus by a very modest 0.5%, then the permanent set after elastic unloading would be 12 le (based on loading to 450 MPa and E 190 GPa). The strain response exhibited stiening for compressive loading (Fig. 5), which prompted unloading when a +10 le oset strain was obtained. During unloading the oset strain decreased below zero, resulting in an apparent permanent set of approximately ÿ20 le. Several other tension and compression cycles were performed on various specimens with similar results. Stiening was observed for loading rates of 1.67 and 6.89 MPa/s. However, the maximum stresses for these two rates were quite dierent, ÿ758 MPa for the slower rate and ÿ414 MPa for the faster rate. In the presence of stiening, the target value was attained for a wide range of stresses, even for the same loading rate. Additionally, we note that Newaz et al. (1996) found the compressive proportional limit of SiC/Ti-15-3 to be 2200 MPa, a value well in excess of any of the maximum compressive stresses that we applied. To investigate the repeatability of stiening, compressive cycling to a speci®ed stress level rather than to a target oset strain was performed. Five compressive loading cycles to ÿ500 MPa were applied. The results from these cycles (shown in Fig. 6) indicate that stiening is not associated with plastic strain since it is present for repeated loadings to a ®xed stress level. Stiening was observed with each loading and the magnitudes of the maximum osets appeared to be random as shown in Fig. 6. There is no apparent relationship
Fig. 6. Oset strains from ®ve consecutive cycles of compression to 500 MPa for SiC/Ti-15-3.
571
between the magnitude of the stiening oset and the cycle number. Further, to a close approximation, the stiening response is nonlinear elastic with the permanent set from each of the ®ve tests being in the range of 2±5 le, i.e., within the resolution of the system, the permanent set is nominally zero. The test setup is susceptible to bending, particularly since the composite specimens were initially warped due to the residual stresses inherent in these materials (Gil et al., 1998). To be sure that stiening is not associated with bending, we measured strains with strain gages bonded to opposite sides of a SiC/Ti-15-3 specimen. In one series of tests the oset strains are determined to be as shown in Fig. 7(a)±(c) for consecutive loadings to maximum compressive stresses of 1034, 690, and 690 MPa, respectively. The results are shown for a second specimen (with similar initial warpage and installed in the same orientation) in Fig. 7(d)±(f) for consecutive loadings to maximum compressive stresses of 552, 690, and 690 MPa, respectively. Each of the three tests shown in Fig. 7(a)±(c) are from the same experimental setup. That is, the specimen was installed in the grips and three tension-compression loading cycles were applied. The same is true for the results shown in Fig. 7(d)± (f). Stiening appears to be a dierent phenomenon than bending because in these tests strain gages on both sides of the specimen indicated stiening. Bending would have caused apparent stiening on only one side of the specimen (Gil et al., 1998). 3.3.2. Elevated temperature Dogbone SiC/Ti-15-3 specimens were also tested at 427°C, 482°C, 538°C and 649°C to determine the in¯uence of temperature. Strains were measured only on one side of the specimen because it was not practical to mount an additional extensometer on the opposite side. Due to the uncertainty about the presence of bending, we are not able to make ®rm statements about whether any observed stiening is related to material behavior or merely an artifact of bending. In the ®rst series of experiments the loading rate was 1.67 MPa/s and unloading (at four times the loading rate) was initiated when the absolute value
572
C.M. Gil et al. / Mechanics of Materials 31 (1999) 565±577
Fig. 7. Oset strains measured by strain gages on the front and back of a SiC/Ti-15-3 specimen; (a)±(c) consecutive cycles on Specimen 6±13, (d)±(f) consecutive cycles on Specimen 6±9.
of the oset strain reached 10 le. As expected, the maximum compressive stress present upon reaching the target value (10 le) decreased as the temperature increased. Temperature (°C) Maximum stress (MPa)
427 482 538 649 517 310 172 103
The oset strains and stresses are shown in Fig. 8(a) and (b), respectively for test temperatures of 427°C, 482°C, 538°C, and 649°C. The oset strains determined for compressive loading at 427°C (Fig. 8(a)) are comparable to those at room temperature that we called stiening. Positive oset strains for compressive loading at 482°C were minimal (not shown) or absent altogether. At 538°C and 649°C, no signi®cant positive oset strains occurred for compressive loading.
The unloading modulus appears to be higher than the loading modulus at 538°C and 649°C (Fig. 8(b)). This could be because the unloading rate was faster than the loading rate. At 427°C, Castelli (1997) has found Ti-15-3 to be slightly strain rate sensitive (i.e., the elastic modulus is a function of the loading rate) and the strain rate sensitivity increases with increasing temperature. The material behavior at 427°C was investigated further by conducting a compression probe using a target value of ÿ10 le, rather than its absolute value, to trigger unloading. That is, a positive oset strain in excess of 10 le would not initiate unloading and a reversal in the oset strain rate followed by a reversal in the sign of the oset strain was possible. Fig. 9 indicates that positive oset strains accumulated to approximately +13 le before reversing and eventually reaching the
C.M. Gil et al. / Mechanics of Materials 31 (1999) 565±577
573
specimen was not removed from the grips prior to heating, we can infer that the reduction or absence of stiening at elevated temperature is true material behavior. 3.4. Inconel 718
Fig. 8. Response of SiC/Ti-15-3 at elevated temperatures; (a) oset strain and (b) stress versus axial strain.
Fig. 9. Oset strains for compression of SiC/Ti-15-3 at 427°C using a target value of ÿ10 le.
target value of ÿ10 le at a compressive stress of ÿ932 MPa. As mentioned previously, it was not practical to measure strain on both sides of the specimen at elevated temperature. However, most specimens were tested at room temperature ®rst, where stiening was observed in compression. The specimens were then heated to the desired test temperature without being removed from the test machine. Gil et al. (1998) indicated that consistent behavior from a specimen is obtained as long as it remains gripped in the test machine. Therefore, since stiening was observed at room temperature and the
To determine whether stiening is associated only with SiC/Ti composites (which are quite complex materials because they are heterogeneous, anisotropic, have residual stresses, and are susceptible to damage) or can occur in homogeneous monolithic materials, we also tested Inconel 718 at room temperature. The loading cycle for each test on a tubular specimen consisted of four uniaxial probes to a target oset strain of 10 le; ®rst tension, then compression, then positive torque, and ®nally negative torque. Only results from the compression probes are shown and discussed because the results of the other three types of probes were routine. The equivalent loading rate was 1.38 MPa/s and the unloading rate was four times faster. To conclusively answer lingering questions about the role of bending and the relation between oset strains and permanent set, strain gages were bonded to all four sides of two IN-718A dogbone specimens. In the results that follow, the four longitudinal strain signals have been averaged together. If we assume that the neutral axis is located at the center of the square cross-section, then this averaging removes the bending strains. Tensile and compressive loading cycles were strain controlled at a rate of 10 le/s. 3.4.1. Solution annealed tube The oset strains for the compression segment of three consecutive cycles of a IN-718S tube at room temperature are shown in Fig. 10(a). As for the type 316 stainless steel, no positive oset strains occurred for compressive loading, suggesting that stiening does not occur in solution annealed materials. The oset strain is irreversible and permanent set is present. The axial strain at which the target oset strain is attained increases slightly from cycle 1 to cycle 3, probably due to the small change in material state that occurs during each probe. This response agrees with our understanding of the deformation of metallic materials.
574
C.M. Gil et al. / Mechanics of Materials 31 (1999) 565±577
the data from the third cycle to verify that the calculated osets were signi®cant with respect to the linearly regressed elastic loading line. The 0.95 con®dence limits were extrapolated to the point where the maximum stiening oset strain occurred. Based on using 97 data points to regress the elastic loading line, r 208; 261 e ÿ 1:973494 MPa;
5
the 0.95 con®dence limits on the elastic strain are [ÿ3,966; ÿ3,960] le at an applied compressive stress of 827 MPa, while the actual measured strain was ÿ3,944 le. The 0.95 con®dence limits are also shown in Fig. 10(b). Clearly, the positive oset strains are well outside these limits indicating that the stiening is statistically signi®cant. 3.4.3. Aged dogbone specimens Tensile and compressive load cycles were applied to two IN-718A dogbone specimens at room temperature. The loading was reversed when the oset strain determined by the control strain gage attained a target value that ranged from 5 to 30 le. However, due to bending, the oset strain determined by a single strain gage was considerably dierent than the average. Table 2 shows the initial load cycles that were applied to each specimen. In Fig. 11, various magnitudes of oset strains (determined from the averaged longitudinal strain
Fig. 10. Oset strains for (a) IN-718S and (b) IN-718A tubular specimens. The maximum stress for each cycle is indicated.
3.4.2. Aged tube The oset strains for the compression segment of three consecutive cycles of a IN-718A tube at room temperature are shown in Fig. 10(b). Positive oset strains, indicating stiening, occur in each cycle. A statistical analysis was performed on
Table 2 Tension and compression cyclic loading of IN-718A dogbone specimens Specimen IN718A-1
Specimen IN718A-2
Cycle
Tension (T) or Compression (C)
Maximum Stress (MPa)
Comments
Cycle
Tension (T) or Compression (C)
Maximum Stress (MPa)
Comments
1 2 3 4 5 6 7 8 9 10 11 12
C C C C C T T T T T T T
ÿ440 ÿ450 ÿ552 ÿ561 ÿ673 272 405 325 520 565 867 734
B B B B D A B B C B D B
1 2 3 4 5 6 7 8 9 10 11 12
T T T T T T T C C C C C
197 188 288 254 356 344 343 ÿ442 ÿ577 ÿ643 ÿ697 ÿ780
A A A A B B B C C D D D
A no oset strains, B oset strains but no permanent set, C oset strains and minimal permanent set, D oset strains and permanent set.
C.M. Gil et al. / Mechanics of Materials 31 (1999) 565±577
575
4. Discussion
Fig. 11. Oset strain as a function of average axial strain for IN-718A dogbone specimens; (a) small osets, (b) intermediate osets, and (c) large osets.
gage signal) are shown. Notice that the tension and compression results shown are from dierent specimens and the maximum oset strain magnitudes increase from Fig. 11(a) to Fig. 11(b) to Fig. 11(c). For tensile loading, the nonlinear elastic and irreversible osets are positive. The 11th cycle (tension) on specimen IN718A-1 (Fig. 11(c)) had a maximum stress of 867 MPa and a permanent set of 92 le. The 12th cycle had a maximum stress of only 734 MPa (Table 2), but still exhibited nonlinear elastic response with no permanent set. For compressive loading, the nonlinear elastic osets are positive and the irreversible osets are negative. Subsequent tests that included a ®ve minute hold time at maximum strain did not exhibit any signi®cant time-dependent response.
Materials consisting of a ductile matrix strengthened by a hard reinforcement phase (continuous SiC ®bers in one case and gamma double prime precipitates in another) have been shown to exhibit the stiening behavior illustrated in Fig. 1, while solution annealed alloys (316 stainless steel and IN-718S) exhibit normal plastic ¯ow behavior. While it is common for polymeric matrix composites to exhibit stiening under tensile loading due to straightening of the carbon ®bers (Diefendorf, 1987), this is clearly not the cause of stiening here. A delay in the transfer of load from the matrix to the reinforcement would cause stiening, but it is unlikely that such a delay would occur in compression and not tension. Although not the topic of this work, these strengthened alloys (SiC/Ti and IN-718A) also exhibit a strength-dierential in tension and compression, that is, their yield and ¯ow stresses are larger in compression than in tension (Gil et al., 1999). The strength-dierential has been investigated for a wide variety of materials (e.g., martensitic steels, metal matrix composites, ceramics, polymers, intermetallics, and soils) and can be attributed to a number of dierent causes; including residual stresses, reinforcement cracking, permanent volume change, and nonlinear elasticity (e.g., Hirth and Cohen, 1970; Drucker, 1973; Arsenault and Wu, 1987; Evans et al., 1991; Doweling, 1999). Only one of the explanations for the strengthdierential also accounts for the observed stiening behavior. Hirth and Cohen (1970) proposed a theory to partially account for the strength-dierential in steels in which dislocations interact with solute atoms causing local distortion of the lattice and leading to local elastic strains that are nonlinear. Hirth and Cohen used a second order stress±strain relation, r e ÿ Ae2 E 1 ÿ 0:24e2
6
where A must be determined, to account for nonlinear elastic interactions between dislocations and interstitial solute atoms. This constitutive equation results in the compressive nonlinear elastic region
576
C.M. Gil et al. / Mechanics of Materials 31 (1999) 565±577
Fig. 12. Nonlinear stress-strain relation of Hirth and Cohen (1970).
having an increasing stiness and the tensile nonlinear elastic region having a decreasing stiness as shown in Fig. 12. This model not only predicts stiening in compression, but also predicts a nonlinear elastic response in tension. The oset strain response shown in Fig. 11 qualitatively agrees with the Hirth and Cohen (1970) nonlinear elastic model. The nonlinear elastic oset strain predicted by Eq. (6) is plotted in Fig. 11. The experimentally determined initial oset strains are in excellent agreement with Eq. (6), but as the oset strain increases due to plastic deformation they diverge from Eq. (6) as expected. The reason for the good agreement may be that the dispersion of precipitates that strengthen the aged Inconel 718 interacts with the dislocations. Our observation that solution annealed Inconel 718, which is not precipitation strengthened and did not exhibit stiening suggests that the presence of a second phase (precipitates in this case) is necessary for stiening to occur. 5. Conclusion Our most important ®nding is that ductile metallic alloys strengthened by hard reinforcements exhibited an increasing stiness under compressive
loading prior to the onset of plastic deformation. The material systems were continuous ®ber composites, SiC/Ti-6242 and SiC/Ti-15-3, as well as the precipitation strengthened nickel base alloy Inconel 718 (aged). As shown in Fig. 11 for aged Inconel 718, stiening under compressive loading is nonlinear elastic. Also shown in Fig. 11 is the fact that the initial tensile response is also nonlinear elastic prior to the onset of plastic deformation, but that the stiness is decreasing rather than increasing. The initial nonlinear elastic response appears to be associated with the constraint provided by a hard second phase and the interaction between it and dislocations. These interactions may make second order terms in the elastic stress±strain relation important (Hirth and Cohen, 1970). The magnitude of the initial nonlinearity is small, but can be very important in yield surface testing where it is not uncommon to use an equivalent oset strain on the order of 10 le as the target value for the yield criterion.
Acknowledgements We thank J.R. Ellis for productive discussions of the experimental results. The ®rst two authors gratefully acknowledge the ®nancial support of the NASA Lewis Research Center (NCC3-481 and NCC3-597).
References Arsenault, R.J., Wu, S.B., 1987. The strength dierential and Bauschinger eects in SiC±Al composites. Mat. Sci. Eng. 96, 77±88. Castelli, M.G., 1997. Private communication. Drucker, D.C., 1973. Plasticity theory, strength-dierential (SD) phenomenon and volume expansion in metals and plastics. Met. Trans. 4, 667±673. Diefendorf, R.J., 1987. Carbon/graphite ®bers, in: Engineered Materials Handbook: Composites, vol. 1. ASM International, Metals Park, pp. 49±53. Doweling, N., 1999. Mechanical Behavior of Materials. Prentice-Hall, Englewood Clis, NJ. Dvorak, G.J., Bahei-El-Din, Y.A., Macheret, Y., Liu, C.H., 1988. An experimental study of elastic±plastic behavior of a ®brous boron±aluminum composite. J. Mech. Phys. Solids 36, 655±687.
C.M. Gil et al. / Mechanics of Materials 31 (1999) 565±577 Ellis, J.R., Bartolotta, P.A., 1997. Adjustable work coil ®xture facilitating the use of induction heating in mechanical testing. In: Kalluri, S., Bonacuse, P.J. (Eds.), Multiaxial Fatigue and Deformation Testing Techniques, ASTM STP 1280. American Society for Testing and Materials, pp. 43± 62. Evans, A.G., Hutchinson, J.W., McMeeking, R.M., 1991. Stress±strain behavior of metal matrix composites with discontinuous reinforcements. Scripta Met. 25, 3±8. Gil, C.M., Lissenden C.J., Lerch B.A., 1998. An investigation of anomalous behavior in metallic-based materials under compressive loading. NASA TM-206640, NASA Lewis Research Center, Cleveland, OH. Gil, C.M., C.J. Lissenden, Lerch B.A., 1999. Determination of yield in Inconel 718 for axial-torsional loading at temperatures up to 649°C. J. Testing & Eval., to appear. Hecker, S.S., 1976. Experimental studies of yield phenomena in biaxially loaded metals. In: Constitutive Equations in Viscoplasticity: Computational and Engineering Aspects. American Society of Mechanical Engineers, New York, pp. 1±33. Helling, D.E., Miller, A.K., Stout, M.G., 1986. An experimental investigation of the yield loci of 1100-0 aluminum, 70:30 brass, and an overaged 2024 aluminum alloy after various prestrains. J. Eng. Mater. Technol. 108, 313±320. Hirth, J.P., Cohen, M., 1970. On the strength-dierential phenomenon in hardened steel. Metall. Trans. 1, 3±8. Khan, A.S., Wang, X., 1993. An experimental study on subsequent yield surface after ®nite shear prestraining. Int. J. Plasticity 9, 889±905. Lissenden, C.J., Lerch, B.A., Ellis, J.R., Robinson, D.N., 1997. Experimental determination of yield and ¯ow surfaces
577
under axial±torsional loading. In: Kalluri, S., Bonacuse, P.J. (Eds.), Multiaxial Fatigue and Deformation Testing Techniques, ASTM STP 1280. American Society for Testing and Materials, pp. 92±112. Liu, K.C., Greenstreet, W.L., 1976. Experimental studies to examine elastic±plastic behavior of metal alloys used in nuclear structures. In: Constitutive Equations in Viscoplasticity: Computational and Engineering Aspects, AMD vol. 20. American Society of Mechanical Engineers, New York, pp. 35±56. Newaz, G.M., Majumdar, B.S., Brust, F.W., 1996, Inelastic deformation mechanisms in SCS-6/Ti-15-3 metal matrix composite (MMC) lamina under compression. In: Johnson, W.S., Larsen, J.M., Cox, B.N. (Eds.) Life Prediction Methodology for Titanium Matrix Composites, ASTM STP 1253. American Society for Testing and Materials, pp. 208±230. Nigam, H., Dvorak, G.J., Bahei-El-Din, Y.A., 1994a. An experimental investigation of elastic±plastic behavior of a ®brous boron±aluminum composite: I. Matrix-dominated mode.. Int. J. Plasticity 10, 23±48. Nigam, H., Dvorak, G.J., Bahei-El-Din, Y.A., 1994b. An experimental investigation of elastic±plastic behavior of a ®brous boron±aluminum composite: II. Fiber-dominated mode. Int. J. Plasticity 10, 49±62. Phillips, A., Liu, C.S., Justusson, J.W., 1972. An experimental investigation of yield surfaces at elevated temperatures. Acta Mechanica 14, 119±146. Wu, H.C., Yeh, W.C., 1991. On the experimental determination of yield surfaces and some results of annealed 304 stainless steel. Int. J. Plasticity 7, 803±826.
www.elsevier.com/locate/mechmat
Unusual nonlinear response of some metallic materials Christopher M. Gil a, Cli J. Lissenden b
a,*
, Bradley A. Lerch
b
a Department of Engineering Science and Mechanics, Pennsylvania State University, University Park, PA 16802, USA Life Prediction Branch, Lewis Research Center, National Aeronautics and Space Administration, Cleveland, OH 44135, USA
Received 28 July 1998; received in revised form 14 April 1999
Abstract An anomalous material response has been observed under the action of compressive loads in ®brous silicon carbide/ titanium (both Ti-6242 and Ti-15-3 alloys) and the nickel-base alloy Inconel 718 in the aged (precipitation hardened) condition. The observed behavior is an increase, rather than a decrease, in the instantaneous Young's modulus with increasing load and is referred to as stiening. This increase is small, but can be signi®cant in yield surface tests where an equivalent oset strain on the order of 10 le (10 ´ 10ÿ6 m/m) is being used. Stiening has been quanti®ed by calculating oset strains from the linear elastic loading line. The oset strains associated with stiening during compressive loading are positive and of the same order as the target oset strains in yield surface tests. Stiening appears to be nonlinear elastic and for SiC/Ti-15-3 decreases as the test temperature increases, suggesting a temperature-dependent mechanism. No stiening occurred in SiC/Ti-15-3 above approximately 500°C. In addition, the tensile response of aged Inconel 718 exhibits a nonlinear elastic response (not stiening) prior to the initiation of plastic deformation. The nonlinear elastic response in both tension and compression agrees with a second order stress±strain relation. The higher order terms may be important due to constraint interactions associated with a strengthening second phase such as precipitates or ®bers. Ó 1999 Elsevier Science Ltd. All rights reserved. Keywords: Yield surface; Plastic ¯ow; Tension test; Compression test; Stiening; Nonlinear elasticity; Metal matrix composite; Precipitation hardened alloy
1. Introduction The experimental work described herein stemmed from an ongoing eort to map out ¯ow surfaces for metallic materials in the axial±shear stress plane at elevated temperatures. A ¯ow surface is similar to a yield surface, except it applies to ratedependent materials. This eort is in support of the development and validation of constitutive models for advanced, structural materials in
* Corresponding author. Tel.: 804-863-5754; fax: 804-8636031; e-mail: [email protected]
aeronautics applications. These experiments involve loading a specimen biaxially up to the onset of yielding and then unloading before signi®cant plastic ¯ow occurs. This is repeated several times on the same specimen using dierent stress ratios until the yield surface has been mapped out, after which the data can be analyzed to obtain ¯ow surfaces. The de®nition of yielding that is used in these experiments in¯uences the resulting yield surface (Khan and Wang, 1993; Hecker, 1976). The de®nition of yielding chosen for this study involves detecting oset strains `on the ¯y'. That is, osets from the linear elastic loading line are determined
0167-6636/99/$ ± see front matter Ó 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 6 3 6 ( 9 9 ) 0 0 0 1 8 - 6
566
C.M. Gil et al. / Mechanics of Materials 31 (1999) 565±577
under load, not after unloading (permanent set). This method greatly expedites detection of yielding, but relies on the assumption that the oset strains are directly related to permanent set. A one-to-one relationship between the maximum oset strain and the permanent set is expected if the initial loading and unloading moduli are equal. By seeking a very small oset strain, on the order of 10 ´ 10ÿ6 m/m (or 10 le), the change in the internal state is minimized and multiple loading excursions (probes) can be conducted on the same specimen to map out the yield surface. Additionally, it removes any concerns about specimen to specimen scatter in the yield surface since the same specimen is used for each probe. It is also cost eective in terms of the number of specimens required. Similar multiprobe single-specimen yield surface experiments have been successfully conducted by many researchers on solution annealed monolithic materials (e.g., Phillips et al., 1972; Liu and Greenstreet, 1976; Helling et al., 1986; Wu and Yeh, 1991) as well as on an annealed boron-aluminum composite (Dvorak et al., 1988; Nigam et al., 1994a, b). Furthermore, the experimental setup used in the current study was implemented by Lissenden et al. (1997) to determine ¯ow surfaces for solution annealed type 316 stainless steel at elevated temperature. However, attempts to determine ¯ow surfaces on a unidirectional SiC/Ti6242 tube have been unsuccessful to date due to an anomalous response, termed stiening, under compressive loading. Stiening is characterized by an increase in the stiness during loading, which creates an oset from the initial loading curve that is often comparable in magnitude, but opposite in sign, to the target value of the strain oset that is sought (Fig. 1). The observation of stiening led to a re-evaluation of the strain measurement apparatus and experimental procedure. After a thorough investigation, the equipment was determined to be functioning properly. In the process of validating the experimental setup and procedures, tubular Inconel 718 specimens were tested. It was found that precipitation hardened (aged) Inconel 718 (IN-718A) exhibited stiening, while solution annealed Inconel 718 (IN-718S) did not.
Fig. 1. Schematic of the Stiening phenomenon in a stress± strain response.
To further study stiening under pure axial loading, thick SiC/Ti-15-3 plates and IN-718A bars were cut into dogbone specimens. Various loading sequences were applied to these specimens. All of the tests described in this paper were aimed at explaining the cause of stiening observed during a yield surface probe.
2. Experimental procedure Quasi-static loading was applied to the various specimens as described below. 2.1. Test equipment Experiments were conducted on one of two biaxial servohydraulic test machines having an axial load capacity of 222,500 N (50,000 lbs) and a torque capacity of 2260 N m (20,000 in lbs). Rotational and translational movements of the actuator are independently controlled by a personal computer coupled with the electronic controller on the machine. The controller allows for rotational control by angle, torque, or shear strain and translational control by displacement, load, or axial strain.
C.M. Gil et al. / Mechanics of Materials 31 (1999) 565±577
Axial and shear strains were measured by an extensometer and in some cases strain gages were also used. The extensometer is a standard, o-theshelf, water-cooled biaxial model with a gage length of 25 mm. Precision strain gages (MicroMeasurements Type EA-06-125BZ-350) having an electrical resistance of 350 X and a gage length of 3.2 mm were installed on select dogbone specimens that were to be tested at room temperature. Four strain gage rosettes were mounted on the type 316 stainless steel tubular specimens (Lissenden et al., 1997). A strain gage conditioner (excitation voltage of 4 V, calibration factor of 1000 le/V) capable of handling multiple strain gages was used. A 5 kW induction heating system with three adjustable coils, described by Ellis and Bartolotta (1997), was used to heat the specimens. Three type K thermocouples were spot welded to the gage section of the specimen, with the center thermocouple used to control the temperature. The two remaining thermocouples were used to ensure a thermal gradient in the gage section of within 1% of the desired test temperature. 2.2. Specimens 2.2.1. Fibrous silicon carbide/titanium One tubular SiC/Ti-6242 specimen was used in this study. The SCS-6 ®bers are oriented along the axis of the tube, which was fabricated using the foil±®ber±foil technique. The exact ®ber volume fraction has not yet been measured, but is nominally 0.25. The tube has inner and outer diameters of 20.7 and 23.3 mm, respectively, and was cut to a length of 203 mm. It was gripped by a ®xture that was designed to transfer stress as uniformly as possible without crushing or splitting the tube. A total of seven [0]32 SiC/Ti-15-3 dogbone specimens were tested in tension and compression. Each specimen contained 35% by volume SCS-6 silicon carbide ®bers aligned longitudinally to reinforce a Ti-15-3 matrix. The specimens were processed by consolidating alternating layers of Ti-15-3 foil and SiC ®bers to achieve a total thickness of approximately 7.6 mm. This thickness was sucient to obviate buckling under the applied compressive loads. The specimens were aged at 700°C for 24 h in a vacuum. Each specimen was
567
203.2 mm long and had the geometry shown in Fig. 2. At each end, stainless steel tabs were bonded to either side of the specimen to ®ll the collet gap in the grips and ensure that the proper gripping pressure was applied. Approximately 44.5 mm of each end of the specimen was gripped. 2.2.2. Monolithics The solution annealed type 316 stainless steel tubular specimen, described in detail by Lissenden et al. (1997), is 229 mm long and has a reduced gage section. The gage section has inner and outer diameters of 22 and 26 mm, respectively. The Inconel 718 tubular specimens are 229 mm long and have inner and outer diameters in the reduced gage section of 15.9 and 21 mm, respectively. After machining, the Inconel-718 specimens were solutioned at 1038°C in argon for 1 h and air cooled. Select specimens were further heat treated as follows: aged at 720°C in argon for 8 h, cooled at 55°C per h to 620°C and held for 8 h, then air cooled to room temperature. Specimens subjected to only the ®rst of these heat treatments are referred to as solutioned (IN-718S), while specimens subjected to both heat treatments are referred to as aged (IN-718A). Two dogbone specimens were machined out of the same Inconel 718 extruded bar stock from which the tubes were machined. The ®nal specimen design was a ¯at, dogbone specimen having a 6.4 ´ 6.4 mm2 gage cross-section as shown in Fig. 2. These specimens were given the same aging treatment described above for the IN-718A tubes. One longitudinal and one transverse strain gage
Fig. 2. Specimen designs for SiC/Ti-15-3 and IN-718A dogbone specimens.
568
C.M. Gil et al. / Mechanics of Materials 31 (1999) 565±577
were mounted at the center of each of the four sides of these two specimens. 2.3. Test procedure Dogbone specimens. Each test was run in axial load and rotation control with the rotation remaining ®xed throughout the test (i.e., a standard uniaxial test), except axial strain control was used for IN-718A specimens. The axial load (engineering stress) was increased monotonically and the initial Young's modulus, E, was determined by linear regression during a prede®ned stress range. A constant data sampling interval was used throughout the test. The stress range was determined by past experience and represented a balance between gathering enough data to accurately calculate the modulus and yet not exceeding the proportional limit. Once the modulus was determined, the strain oset from the linear elastic loading line, eoff , was calculated from the total strain, e, the applied stress, r, the initial stress, r0 , and the modulus, r ÿ r0 1 eoff e ÿ E and continually compared to a target value (usually 10 le). When the oset strain reached the target value the specimen was unloaded at a rate four times that of the loading rate, except an unloading rate equal to the loading rate was used for IN-718A specimens. The increased unloading rate was used to expedite the tests since several repeats of each test were commonly performed. This type of test is typical of what is done in the mapping of a yield or ¯ow surface (Lissenden et al., 1997). The primary loading sequence was tension until the oset strain reached a target value, followed by compression until the oset strain reached a target value. Results are presented for IN-718A specimens tested at room temperature, in which both strain gage and extensometer data were collected. Tests on SiC/Ti-15-3 at 427°C, 482°C, 538°C and 649°C are the only elevated temperature results that are reported in this paper. No strain gages were used at elevated temperature. Tubular specimens. Proportional axial±torsional loads were applied to the type 316 stainless steel
tube and the SiC/Ti-6242 tube using the procedure described in detail by Lissenden et al. (1997). We present results from the Inconel 718 tubular specimens for uniaxial compression only. For the current report the nature of the biaxial loading is unimportant and will not be discussed further. The axial oset strain was determined using Eq. (1). 3. Results Experimental results for dierent materials will be presented in the order in which the tests were conducted: 1. a type 316 stainless steel tubular specimen, where the material stress±strain response was as expected, 2. a SiC/Ti-6242 tubular specimen, where anomalous material stress±strain response was ®rst detected, 3. SiC/Ti-15-3 dogbone specimens, where the anomalous stress±strain response was studied in detail, and which represent the bulk of the results presented in this report, 4. IN-718S (no stiening) and IN-718A (stiening) tubular specimens, and 5. IN-718A dogbone specimens that exhibited nonlinear elastic behavior in both tension and compression and represent the most informative results. 3.1. Type 316 stainless steel Results from yield surface experiments on solution annealed 316 stainless steel are presented by Lissenden et al. (1997). Here we will only look at the axial stress±strain response from one typical probe of a yield surface at room temperature (having an eective loading rate of 1 MPa/s). The axial responses from other probes, including nearly pure uniaxial compression, were similar. Strains were measured during proportional loading probing experiments with the extensometer and strain gages. The axial stress±strain response from the ®rst probe (a 12° probe angle in the axial± shear stress plane; that is, predominantly axial tension) of surface determination run number six is shown in Fig. 3; shear stresses (s 0.2126r) and
C.M. Gil et al. / Mechanics of Materials 31 (1999) 565±577
569
and strain gage measured strains after the load reversal (Fig. 3) was generally not observed in the oset shear strain data. 3.2. SiC/Ti-6242 tubular specimens
Fig. 3. Solution annealed type 316 stainless steel axial response to a yield surface probe at 12° in the axial±shear stress plane.
strains were also present but are not shown here. The initial response is linear elastic with the linearly regressed stress±strain equations in the strain range 0 < e < 150 le being r 196; 510e 0:11652 extensometer r 190; 970e 0:13285 strain gages;
2
with r in MPa and e in m/m. These equations represent the elastic loading lines from the extensometer and strain gage data, respectively. The stress±strain curve and linear ®t in Fig. 3 are based on the extensometer strain measurement. Fig. 3 also shows the (axial) oset strain as a function of (axial) total strain for both the extensometer and strain gages. The oset strain starts to increase at a total strain of 400 le and once the load is removed the oset strain remains as permanent set. This is typical of plastic deformation in metals and strongly suggests that the oset strain is an excellent indicator of the permanent set. Notice that while the equations of the elastic loading lines were dierent for the extensometer and the strain gages (that is, a 2.8% dierence in Young's modulus), the oset strains measured were nearly identical. Although not shown, for each yield surface probe, each oset strain component had the same sign as its total strain counterpart (excluding scatter). We note that the good agreement between extensometer
Having good success with the type 316 SS specimens, we attempted to conduct yield surface tests on SiC/Ti-6242 tubular specimens at room temperature. Fig. 4 shows the axial stress and the axial oset strain as a function of the total axial strain for a probe angle of ÿ168° (predominantly axial compression, s ÿ0.2126r) and an equivalent loading rate of 2.1 MPa/s. Again, shear stresses and strains were present, but are not shown. Since we had anticipated the center of the ¯ow surface to be shifted in the compressive stress direction due to residual stresses, each probe was started at a compressive stress of approximately 420 MPa. The elastic loading line in the strain range ÿ2500 < e < ÿ2200 le is r 189; 612 e 1:2907 MPa
3
and although the stress±strain response appears to be linear throughout, small oset strains are present. Interestingly, these oset strains have positive, rather than negative, magnitudes and therefore indicate stiening. Additionally, the oset strain decreased during unloading more than it
Fig. 4. SiC/Ti-6242 axial response to a yield surface probe at ÿ168° in the axial±shear stress plane.
570
C.M. Gil et al. / Mechanics of Materials 31 (1999) 565±577
had increased during loading, resulting in a very small (3 le) compressive permanent set. While the stiening observed during loading is very representative of what we observed for compressive loading, and even occasionally for torsional loading, the calculated oset strains during unloading were somewhat erratic. Even though these oset strains are small, they are statistically meaningful. We have determined the 95% con®dence limits on the linearly regressed loading line and extrapolated them up to the maximum load. Doing so, we found that the measured stress±strain response falls above the upper limit. There were 250 data points used in the regression and the correlation factor was 0.99958. In addition, this strain range was very successful for determining the elastic loading line during tensile loading. 3.3. SiC/Ti-15-3 dogbone specimens SiC/Ti-15-3 dogbone specimens, rather than the few available SiC/Ti-6242 tubular specimens, were used to closely examine uniaxial response. These specimens have the same SCS-6 ®ber and a weak interface similar to the SiC/Ti-6242 specimens. Hence, the behavior of the dogbone specimens should be similar to that of the tubes. In addition, the SiC/Ti-15-3 specimens were plentiful and less expensive than the tubes. Unfortunately, these specimens were not entirely ¯at, which complicates the interpretation of the results. 3.3.1. Room temperature Tensile and compressive loadings were applied at room temperature at a rate of 1.67 MPa/s. Unloading was triggered by the oset strain attaining a target value of 10 le in this and all subsequent tests, unless otherwise noted. Additionally, reported strains were measured by the extensometer unless stated to the contrary. The coecients of the elastic loading line, r Ee r0 ;
Table 1 SiC/Ti-15-3 elastic loading line coecients for tensile and compressive loading Test 1 2 3 4
Tension
Compression
E (MPa)
r0 (MPa)
E (MPa)
r0 (MPa)
190,565 189,369 189,151 189,092
0.104337 0.047183 0.721945 0.439698
191,351 190,984 191,216 190,865
ÿ1.78900 ÿ3.85694 ÿ3.37030 ÿ4.08399
the compressive Young's modulus is approximately 1% larger than the tensile Young's modulus in these four tests. The larger (negative) initial stresses for compressive loading are an indication of the small permanent set that occurred due to the tensile loading. The typical calculated oset strain is shown in Fig. 5 as a function of the total strain for tensile and compressive loadings. The oset strain is essentially zero during elastic loading (e< |1000| le) and is positive thereafter. The tensile (compressive) stress present at the load reversal was 451 MPa (ÿ601 MPa). The permanent set turned out to be more than twice the maximum oset during loading. This dierence could be due to measurement problems with the extensometer upon reversal of the load (such as reseating of the extensometer rods in their indents) or due to ratedependence of Young's modulus (a viscoelastic eect) since unloading was done at four times the loading rate. We have no data indicating that Young's modulus is loading rate-dependent at room temperature, but it does appear to be ratedependent at elevated temperature. However, even a very slight rate-dependence can have an eect. If
4
are shown in Table 1 for four consecutive tests on the same specimen. There is a 1% variation in the tensile Young's modulus and a 0.2% variation in the compressive Young's modulus. Additionally,
Fig. 5. Oset strains from both tensile and compressive loadings of SiC/Ti-15-3.
C.M. Gil et al. / Mechanics of Materials 31 (1999) 565±577
the faster unloading rate were to increase the Young's modulus by a very modest 0.5%, then the permanent set after elastic unloading would be 12 le (based on loading to 450 MPa and E 190 GPa). The strain response exhibited stiening for compressive loading (Fig. 5), which prompted unloading when a +10 le oset strain was obtained. During unloading the oset strain decreased below zero, resulting in an apparent permanent set of approximately ÿ20 le. Several other tension and compression cycles were performed on various specimens with similar results. Stiening was observed for loading rates of 1.67 and 6.89 MPa/s. However, the maximum stresses for these two rates were quite dierent, ÿ758 MPa for the slower rate and ÿ414 MPa for the faster rate. In the presence of stiening, the target value was attained for a wide range of stresses, even for the same loading rate. Additionally, we note that Newaz et al. (1996) found the compressive proportional limit of SiC/Ti-15-3 to be 2200 MPa, a value well in excess of any of the maximum compressive stresses that we applied. To investigate the repeatability of stiening, compressive cycling to a speci®ed stress level rather than to a target oset strain was performed. Five compressive loading cycles to ÿ500 MPa were applied. The results from these cycles (shown in Fig. 6) indicate that stiening is not associated with plastic strain since it is present for repeated loadings to a ®xed stress level. Stiening was observed with each loading and the magnitudes of the maximum osets appeared to be random as shown in Fig. 6. There is no apparent relationship
Fig. 6. Oset strains from ®ve consecutive cycles of compression to 500 MPa for SiC/Ti-15-3.
571
between the magnitude of the stiening oset and the cycle number. Further, to a close approximation, the stiening response is nonlinear elastic with the permanent set from each of the ®ve tests being in the range of 2±5 le, i.e., within the resolution of the system, the permanent set is nominally zero. The test setup is susceptible to bending, particularly since the composite specimens were initially warped due to the residual stresses inherent in these materials (Gil et al., 1998). To be sure that stiening is not associated with bending, we measured strains with strain gages bonded to opposite sides of a SiC/Ti-15-3 specimen. In one series of tests the oset strains are determined to be as shown in Fig. 7(a)±(c) for consecutive loadings to maximum compressive stresses of 1034, 690, and 690 MPa, respectively. The results are shown for a second specimen (with similar initial warpage and installed in the same orientation) in Fig. 7(d)±(f) for consecutive loadings to maximum compressive stresses of 552, 690, and 690 MPa, respectively. Each of the three tests shown in Fig. 7(a)±(c) are from the same experimental setup. That is, the specimen was installed in the grips and three tension-compression loading cycles were applied. The same is true for the results shown in Fig. 7(d)± (f). Stiening appears to be a dierent phenomenon than bending because in these tests strain gages on both sides of the specimen indicated stiening. Bending would have caused apparent stiening on only one side of the specimen (Gil et al., 1998). 3.3.2. Elevated temperature Dogbone SiC/Ti-15-3 specimens were also tested at 427°C, 482°C, 538°C and 649°C to determine the in¯uence of temperature. Strains were measured only on one side of the specimen because it was not practical to mount an additional extensometer on the opposite side. Due to the uncertainty about the presence of bending, we are not able to make ®rm statements about whether any observed stiening is related to material behavior or merely an artifact of bending. In the ®rst series of experiments the loading rate was 1.67 MPa/s and unloading (at four times the loading rate) was initiated when the absolute value
572
C.M. Gil et al. / Mechanics of Materials 31 (1999) 565±577
Fig. 7. Oset strains measured by strain gages on the front and back of a SiC/Ti-15-3 specimen; (a)±(c) consecutive cycles on Specimen 6±13, (d)±(f) consecutive cycles on Specimen 6±9.
of the oset strain reached 10 le. As expected, the maximum compressive stress present upon reaching the target value (10 le) decreased as the temperature increased. Temperature (°C) Maximum stress (MPa)
427 482 538 649 517 310 172 103
The oset strains and stresses are shown in Fig. 8(a) and (b), respectively for test temperatures of 427°C, 482°C, 538°C, and 649°C. The oset strains determined for compressive loading at 427°C (Fig. 8(a)) are comparable to those at room temperature that we called stiening. Positive oset strains for compressive loading at 482°C were minimal (not shown) or absent altogether. At 538°C and 649°C, no signi®cant positive oset strains occurred for compressive loading.
The unloading modulus appears to be higher than the loading modulus at 538°C and 649°C (Fig. 8(b)). This could be because the unloading rate was faster than the loading rate. At 427°C, Castelli (1997) has found Ti-15-3 to be slightly strain rate sensitive (i.e., the elastic modulus is a function of the loading rate) and the strain rate sensitivity increases with increasing temperature. The material behavior at 427°C was investigated further by conducting a compression probe using a target value of ÿ10 le, rather than its absolute value, to trigger unloading. That is, a positive oset strain in excess of 10 le would not initiate unloading and a reversal in the oset strain rate followed by a reversal in the sign of the oset strain was possible. Fig. 9 indicates that positive oset strains accumulated to approximately +13 le before reversing and eventually reaching the
C.M. Gil et al. / Mechanics of Materials 31 (1999) 565±577
573
specimen was not removed from the grips prior to heating, we can infer that the reduction or absence of stiening at elevated temperature is true material behavior. 3.4. Inconel 718
Fig. 8. Response of SiC/Ti-15-3 at elevated temperatures; (a) oset strain and (b) stress versus axial strain.
Fig. 9. Oset strains for compression of SiC/Ti-15-3 at 427°C using a target value of ÿ10 le.
target value of ÿ10 le at a compressive stress of ÿ932 MPa. As mentioned previously, it was not practical to measure strain on both sides of the specimen at elevated temperature. However, most specimens were tested at room temperature ®rst, where stiening was observed in compression. The specimens were then heated to the desired test temperature without being removed from the test machine. Gil et al. (1998) indicated that consistent behavior from a specimen is obtained as long as it remains gripped in the test machine. Therefore, since stiening was observed at room temperature and the
To determine whether stiening is associated only with SiC/Ti composites (which are quite complex materials because they are heterogeneous, anisotropic, have residual stresses, and are susceptible to damage) or can occur in homogeneous monolithic materials, we also tested Inconel 718 at room temperature. The loading cycle for each test on a tubular specimen consisted of four uniaxial probes to a target oset strain of 10 le; ®rst tension, then compression, then positive torque, and ®nally negative torque. Only results from the compression probes are shown and discussed because the results of the other three types of probes were routine. The equivalent loading rate was 1.38 MPa/s and the unloading rate was four times faster. To conclusively answer lingering questions about the role of bending and the relation between oset strains and permanent set, strain gages were bonded to all four sides of two IN-718A dogbone specimens. In the results that follow, the four longitudinal strain signals have been averaged together. If we assume that the neutral axis is located at the center of the square cross-section, then this averaging removes the bending strains. Tensile and compressive loading cycles were strain controlled at a rate of 10 le/s. 3.4.1. Solution annealed tube The oset strains for the compression segment of three consecutive cycles of a IN-718S tube at room temperature are shown in Fig. 10(a). As for the type 316 stainless steel, no positive oset strains occurred for compressive loading, suggesting that stiening does not occur in solution annealed materials. The oset strain is irreversible and permanent set is present. The axial strain at which the target oset strain is attained increases slightly from cycle 1 to cycle 3, probably due to the small change in material state that occurs during each probe. This response agrees with our understanding of the deformation of metallic materials.
574
C.M. Gil et al. / Mechanics of Materials 31 (1999) 565±577
the data from the third cycle to verify that the calculated osets were signi®cant with respect to the linearly regressed elastic loading line. The 0.95 con®dence limits were extrapolated to the point where the maximum stiening oset strain occurred. Based on using 97 data points to regress the elastic loading line, r 208; 261 e ÿ 1:973494 MPa;
5
the 0.95 con®dence limits on the elastic strain are [ÿ3,966; ÿ3,960] le at an applied compressive stress of 827 MPa, while the actual measured strain was ÿ3,944 le. The 0.95 con®dence limits are also shown in Fig. 10(b). Clearly, the positive oset strains are well outside these limits indicating that the stiening is statistically signi®cant. 3.4.3. Aged dogbone specimens Tensile and compressive load cycles were applied to two IN-718A dogbone specimens at room temperature. The loading was reversed when the oset strain determined by the control strain gage attained a target value that ranged from 5 to 30 le. However, due to bending, the oset strain determined by a single strain gage was considerably dierent than the average. Table 2 shows the initial load cycles that were applied to each specimen. In Fig. 11, various magnitudes of oset strains (determined from the averaged longitudinal strain
Fig. 10. Oset strains for (a) IN-718S and (b) IN-718A tubular specimens. The maximum stress for each cycle is indicated.
3.4.2. Aged tube The oset strains for the compression segment of three consecutive cycles of a IN-718A tube at room temperature are shown in Fig. 10(b). Positive oset strains, indicating stiening, occur in each cycle. A statistical analysis was performed on
Table 2 Tension and compression cyclic loading of IN-718A dogbone specimens Specimen IN718A-1
Specimen IN718A-2
Cycle
Tension (T) or Compression (C)
Maximum Stress (MPa)
Comments
Cycle
Tension (T) or Compression (C)
Maximum Stress (MPa)
Comments
1 2 3 4 5 6 7 8 9 10 11 12
C C C C C T T T T T T T
ÿ440 ÿ450 ÿ552 ÿ561 ÿ673 272 405 325 520 565 867 734
B B B B D A B B C B D B
1 2 3 4 5 6 7 8 9 10 11 12
T T T T T T T C C C C C
197 188 288 254 356 344 343 ÿ442 ÿ577 ÿ643 ÿ697 ÿ780
A A A A B B B C C D D D
A no oset strains, B oset strains but no permanent set, C oset strains and minimal permanent set, D oset strains and permanent set.
C.M. Gil et al. / Mechanics of Materials 31 (1999) 565±577
575
4. Discussion
Fig. 11. Oset strain as a function of average axial strain for IN-718A dogbone specimens; (a) small osets, (b) intermediate osets, and (c) large osets.
gage signal) are shown. Notice that the tension and compression results shown are from dierent specimens and the maximum oset strain magnitudes increase from Fig. 11(a) to Fig. 11(b) to Fig. 11(c). For tensile loading, the nonlinear elastic and irreversible osets are positive. The 11th cycle (tension) on specimen IN718A-1 (Fig. 11(c)) had a maximum stress of 867 MPa and a permanent set of 92 le. The 12th cycle had a maximum stress of only 734 MPa (Table 2), but still exhibited nonlinear elastic response with no permanent set. For compressive loading, the nonlinear elastic osets are positive and the irreversible osets are negative. Subsequent tests that included a ®ve minute hold time at maximum strain did not exhibit any signi®cant time-dependent response.
Materials consisting of a ductile matrix strengthened by a hard reinforcement phase (continuous SiC ®bers in one case and gamma double prime precipitates in another) have been shown to exhibit the stiening behavior illustrated in Fig. 1, while solution annealed alloys (316 stainless steel and IN-718S) exhibit normal plastic ¯ow behavior. While it is common for polymeric matrix composites to exhibit stiening under tensile loading due to straightening of the carbon ®bers (Diefendorf, 1987), this is clearly not the cause of stiening here. A delay in the transfer of load from the matrix to the reinforcement would cause stiening, but it is unlikely that such a delay would occur in compression and not tension. Although not the topic of this work, these strengthened alloys (SiC/Ti and IN-718A) also exhibit a strength-dierential in tension and compression, that is, their yield and ¯ow stresses are larger in compression than in tension (Gil et al., 1999). The strength-dierential has been investigated for a wide variety of materials (e.g., martensitic steels, metal matrix composites, ceramics, polymers, intermetallics, and soils) and can be attributed to a number of dierent causes; including residual stresses, reinforcement cracking, permanent volume change, and nonlinear elasticity (e.g., Hirth and Cohen, 1970; Drucker, 1973; Arsenault and Wu, 1987; Evans et al., 1991; Doweling, 1999). Only one of the explanations for the strengthdierential also accounts for the observed stiening behavior. Hirth and Cohen (1970) proposed a theory to partially account for the strength-dierential in steels in which dislocations interact with solute atoms causing local distortion of the lattice and leading to local elastic strains that are nonlinear. Hirth and Cohen used a second order stress±strain relation, r e ÿ Ae2 E 1 ÿ 0:24e2
6
where A must be determined, to account for nonlinear elastic interactions between dislocations and interstitial solute atoms. This constitutive equation results in the compressive nonlinear elastic region
576
C.M. Gil et al. / Mechanics of Materials 31 (1999) 565±577
Fig. 12. Nonlinear stress-strain relation of Hirth and Cohen (1970).
having an increasing stiness and the tensile nonlinear elastic region having a decreasing stiness as shown in Fig. 12. This model not only predicts stiening in compression, but also predicts a nonlinear elastic response in tension. The oset strain response shown in Fig. 11 qualitatively agrees with the Hirth and Cohen (1970) nonlinear elastic model. The nonlinear elastic oset strain predicted by Eq. (6) is plotted in Fig. 11. The experimentally determined initial oset strains are in excellent agreement with Eq. (6), but as the oset strain increases due to plastic deformation they diverge from Eq. (6) as expected. The reason for the good agreement may be that the dispersion of precipitates that strengthen the aged Inconel 718 interacts with the dislocations. Our observation that solution annealed Inconel 718, which is not precipitation strengthened and did not exhibit stiening suggests that the presence of a second phase (precipitates in this case) is necessary for stiening to occur. 5. Conclusion Our most important ®nding is that ductile metallic alloys strengthened by hard reinforcements exhibited an increasing stiness under compressive
loading prior to the onset of plastic deformation. The material systems were continuous ®ber composites, SiC/Ti-6242 and SiC/Ti-15-3, as well as the precipitation strengthened nickel base alloy Inconel 718 (aged). As shown in Fig. 11 for aged Inconel 718, stiening under compressive loading is nonlinear elastic. Also shown in Fig. 11 is the fact that the initial tensile response is also nonlinear elastic prior to the onset of plastic deformation, but that the stiness is decreasing rather than increasing. The initial nonlinear elastic response appears to be associated with the constraint provided by a hard second phase and the interaction between it and dislocations. These interactions may make second order terms in the elastic stress±strain relation important (Hirth and Cohen, 1970). The magnitude of the initial nonlinearity is small, but can be very important in yield surface testing where it is not uncommon to use an equivalent oset strain on the order of 10 le as the target value for the yield criterion.
Acknowledgements We thank J.R. Ellis for productive discussions of the experimental results. The ®rst two authors gratefully acknowledge the ®nancial support of the NASA Lewis Research Center (NCC3-481 and NCC3-597).
References Arsenault, R.J., Wu, S.B., 1987. The strength dierential and Bauschinger eects in SiC±Al composites. Mat. Sci. Eng. 96, 77±88. Castelli, M.G., 1997. Private communication. Drucker, D.C., 1973. Plasticity theory, strength-dierential (SD) phenomenon and volume expansion in metals and plastics. Met. Trans. 4, 667±673. Diefendorf, R.J., 1987. Carbon/graphite ®bers, in: Engineered Materials Handbook: Composites, vol. 1. ASM International, Metals Park, pp. 49±53. Doweling, N., 1999. Mechanical Behavior of Materials. Prentice-Hall, Englewood Clis, NJ. Dvorak, G.J., Bahei-El-Din, Y.A., Macheret, Y., Liu, C.H., 1988. An experimental study of elastic±plastic behavior of a ®brous boron±aluminum composite. J. Mech. Phys. Solids 36, 655±687.
C.M. Gil et al. / Mechanics of Materials 31 (1999) 565±577 Ellis, J.R., Bartolotta, P.A., 1997. Adjustable work coil ®xture facilitating the use of induction heating in mechanical testing. In: Kalluri, S., Bonacuse, P.J. (Eds.), Multiaxial Fatigue and Deformation Testing Techniques, ASTM STP 1280. American Society for Testing and Materials, pp. 43± 62. Evans, A.G., Hutchinson, J.W., McMeeking, R.M., 1991. Stress±strain behavior of metal matrix composites with discontinuous reinforcements. Scripta Met. 25, 3±8. Gil, C.M., Lissenden C.J., Lerch B.A., 1998. An investigation of anomalous behavior in metallic-based materials under compressive loading. NASA TM-206640, NASA Lewis Research Center, Cleveland, OH. Gil, C.M., C.J. Lissenden, Lerch B.A., 1999. Determination of yield in Inconel 718 for axial-torsional loading at temperatures up to 649°C. J. Testing & Eval., to appear. Hecker, S.S., 1976. Experimental studies of yield phenomena in biaxially loaded metals. In: Constitutive Equations in Viscoplasticity: Computational and Engineering Aspects. American Society of Mechanical Engineers, New York, pp. 1±33. Helling, D.E., Miller, A.K., Stout, M.G., 1986. An experimental investigation of the yield loci of 1100-0 aluminum, 70:30 brass, and an overaged 2024 aluminum alloy after various prestrains. J. Eng. Mater. Technol. 108, 313±320. Hirth, J.P., Cohen, M., 1970. On the strength-dierential phenomenon in hardened steel. Metall. Trans. 1, 3±8. Khan, A.S., Wang, X., 1993. An experimental study on subsequent yield surface after ®nite shear prestraining. Int. J. Plasticity 9, 889±905. Lissenden, C.J., Lerch, B.A., Ellis, J.R., Robinson, D.N., 1997. Experimental determination of yield and ¯ow surfaces
577
under axial±torsional loading. In: Kalluri, S., Bonacuse, P.J. (Eds.), Multiaxial Fatigue and Deformation Testing Techniques, ASTM STP 1280. American Society for Testing and Materials, pp. 92±112. Liu, K.C., Greenstreet, W.L., 1976. Experimental studies to examine elastic±plastic behavior of metal alloys used in nuclear structures. In: Constitutive Equations in Viscoplasticity: Computational and Engineering Aspects, AMD vol. 20. American Society of Mechanical Engineers, New York, pp. 35±56. Newaz, G.M., Majumdar, B.S., Brust, F.W., 1996, Inelastic deformation mechanisms in SCS-6/Ti-15-3 metal matrix composite (MMC) lamina under compression. In: Johnson, W.S., Larsen, J.M., Cox, B.N. (Eds.) Life Prediction Methodology for Titanium Matrix Composites, ASTM STP 1253. American Society for Testing and Materials, pp. 208±230. Nigam, H., Dvorak, G.J., Bahei-El-Din, Y.A., 1994a. An experimental investigation of elastic±plastic behavior of a ®brous boron±aluminum composite: I. Matrix-dominated mode.. Int. J. Plasticity 10, 23±48. Nigam, H., Dvorak, G.J., Bahei-El-Din, Y.A., 1994b. An experimental investigation of elastic±plastic behavior of a ®brous boron±aluminum composite: II. Fiber-dominated mode. Int. J. Plasticity 10, 49±62. Phillips, A., Liu, C.S., Justusson, J.W., 1972. An experimental investigation of yield surfaces at elevated temperatures. Acta Mechanica 14, 119±146. Wu, H.C., Yeh, W.C., 1991. On the experimental determination of yield surfaces and some results of annealed 304 stainless steel. Int. J. Plasticity 7, 803±826.