Читать книгу Engineering Physics of High-Temperature Materials - Nirmal K. Sinha - Страница 37

Let us very briefly look at the essence of the technique of looking backward. The principle of the use of hindsight for a creep test is illustrated in Figure 1.4 using the traditional presentation of linear timescale. In this case of a popular nickel‐base superalloy (Waspaloy), a tensile specimen is first loaded fully and rapidly (rise time <1 s) and unloaded completely (also in <1 s) after a creep time of 200 s, well within the transient or primary creep range, in comparison with 800 s for the time to reach the minimum creep rate for this level of stress. The axial strain recovery, after rapid removal of the load, was monitored for a relatively long time until a permanent or a viscous strain could be evaluated. Then, the same specimen was loaded for a long time to the accelerating tertiary stage and then fully unloaded, and strain recovery was recorded. For clarity, the strain level and time for the mcr for the longer‐term test are also shown. The mcr was determined from the strain rate versus time curve. The differences in the amount of permanent or viscous strain for the two tests are clearly noticeable. As expected, the permanent or viscous strain that occurred during the long test (duration of 2341 s) is significantly greater than that of the short test (for 200 s). Note the amount of elastic strain, delayed elastic strain, and viscous strain, shown for the longer‐term test. The delayed elastic strain is small, but not negligible. Similar observations were also noticed for the short‐time test. These observations provide clear indication that delayed elasticity, mostly ignored so far in high‐temperature rheological models emphasizing only mcr or steady‐state creep rate, should be given due consideration in order to get a better understanding of the mechanics of high‐temperature creep and failure. It should be mentioned here that the time span for full load application or the rise time of less than 1 s is also expected to minimize the scatter in the stress dependency of mcr as discussed by Bressers et al. (1981).

Figure 1.4 Short‐term (200 s) and longer‐term (2341 s) tensile SRRTs on a single specimen of polycrystalline nickel‐base superalloy Waspaloy at 1005 K (732 °C) for 650 MPa using linear timescale.

Source: N. K. Sinha.

There are innumerable sets of creep curves for a wide variety of manufactured and natural materials illustrating transient and tertiary creep stages, but the recovery on full unloading is rarely reported. There are, however, examples of stress‐dip tests in which creep continues after a short recovery on partial unloading during the steady state or actually at mcr that occurs at evolved microstructure corresponding to this state. Unfortunately, stress‐dip tests do not provide useful information on transient creep at the beginning of a creep test and the characteristics of neither the delayed elastic deformation nor the viscous flow corresponding to the original, undeformed and undamaged microstructure.

Figure 1.4 exemplifies a unique set of results for a complex nickel‐based aerospace alloy. It brings out the fact that the delayed elastic strain, ε _d, recovered on full unloading well within the tertiary stage of creep, after mcr, was not negligible and not “absorbed” within the viscous component. The long‐term test (2341 s) is noticeably larger than that of the 200 s test. Hence, ε _d increases with time. This raises the question as to the mechanism(s) responsible for generating delayed elasticity in polycrystalline materials that may have far‐reaching consequences, presented in Chapter 5, in developing physically based creep models.

Conventionally, a typical (often called normal) creep curve is described in terms of three stages: a primary regime during which the creep rate continuously decreases, a secondary regime where the creep rate is at a minimum, and a tertiary regime where the creep rate continuously increases, leading to rupture. All these three stages (except for the rupture) can be seen in the long‐term creep curve in Figure 1.5. The slope of the curve at a given time gives the creep rate at that time. It varies with time. However, the creep rate at a given time provides “total strain rate” or the rate of the “sum of reversible and irreversible strain” at that time. If the creep rate is plotted against time on a log–log scale, a typical creep response can be described by only two regimes: primary and tertiary. The minimum creep rate represents only the transition point between these two regimes. From the physics point of view, its numerical value provides no information either during the primary regime or the tertiary stage – the regimes important for engineering designs and life cycle management.

The use of the linear scale for time obscures the initial conditions of CL creep tests. Moreover, traditional methods of using a dead‐load lever system, although very simple and ideal for conducting very long‐term creep‐rupture tests with durations of 10 000 hours or more (e.g. Holdsworth et al. 2005; Yagi 2005; Kimura et al. 2009), do not allow the load to be applied as quickly as possible to satisfy the common assumption of “instantaneous” load application. Consequently, the materials science literature is full of the reported amount of initial strain on loading, ε _ο. The use of linear timescale conveniently “hides” or obscures the weakness of the initial experimental conditions, albeit unavoidable for dead‐load systems. Nonetheless, a lot of undue emphasis has been put on the numerical values of ε _ο. There is nothing fundamental about this initial strain. It simply depends on the test and data logging system and, to a greater degree, on the manual dexterity of an experimentalist. Figure 1.5 presents the results of Figure 1.4 using a logarithmic scale for time. Note the load‐application times of less than 1 s for both tests. This has been possible only by using a computer‐controlled closed‐loop servo‐hydraulic test system (described in Chapter 4).

Figure 1.5 Results of Figure 1.4 shown in a logarithmic timescale, illustrating load‐application times (<1 s) and similarities between the minimum creep rate of 2.85 × 10⁻⁶ s⁻¹ and the average viscous strain rates of 3.05 × 10⁻⁶ s⁻¹ and 2.68 × 10⁻⁶ s⁻¹, respectively, for 200 s and 2432 s from the corresponding, ε_v, at the end of recovery.

Source: N. K. Sinha.

Figure 1.5 also shows the “average viscous strain rate”, , during the creep time or the “strain relaxation” time, t _SR, of 200 s and 2432 s, respectively, given by

(1.1)

where ε_v is the “measured permanent or viscous strain” after full recovery, as shown in Figure 1.5.

A set of five long‐term SRRT curves for Waspaloy is presented in Figure 1.6. All of these tests were carried out to periods well within the tertiary stages, but unloaded completely and extremely rapidly before fracturing and the strains were monitored for a long period to explore strain recoveries. Since the total strain even at unloading times was less than 1.2%, the true stress increased only slightly even at the time of unloading. These results are consistent with numerous published creep data in the literature, but with two important differences. These results present, for the first time to the authors’ knowledge, the systematic observations on creep recovery after full unloading during the tertiary stages and specimen response during “rise times” to apply the full load. Figure 1.6 clearly illustrates the strain–time response during the “rise time” (from which elastic modulus E can be determined for each of the specimens). It also shows the creep curves with the corresponding minimum creep rates, and most importantly the permanent viscous strains, ε _v, and the corresponding recovered delayed elastic strains, ε _d (des), at the end of the tests. Analysis of the short‐term and long‐term SRRT data (Figure 1.6) indicated that the creep strain at minimum creep rate consists of a significant amount of recoverable delayed elastic strain (32% at 450 MPa and 38% at 650 MPa).

Figure 1.6 Strain–time curves, showing rise time to apply full load (<1 s), during creep for five SRRTs, including the longer duration results of Figure 1.4, on a polycrystalline nickel‐base superalloy Waspaloy at 1005 K (732 °C) for initial stresses 450–650 MPa. Note the delayed elastic recovery and viscous strains after unloading.

Source: N. K. Sinha.

Figure 1.6 points out that the time to mcr, t _m , decreased significantly, over an order of magnitude, with increase in engineering stress from 450 to 650 MPa. The dependence of t _m on stress, σ, may be expressed by the following relationship:

(1.2)

where M = 1.61 × 10²⁹ and p = 9.36. Such a relationship has commonly been seen for the dependence of rupture or fracture time on stress, leading to popular ideas on relating mcr to failures at high temperatures (see Chapter 6). Note that the total strain, ε _min (which includes the elastic strain, ε _e), corresponding to mcr increased only from 0.38% for 450 MPa to about 0.64% for 650 MPa. The corresponding values of the creep strain (ε _min−ε _e) increased from 0.12 to 0.27%. This type of diminished stress dependency of ε _min agrees with general observations available in the literature on the relatively small increase in strain at mcr with increase in stress. In fact, the elongation at fracture hardly varies with stress. However, note the increase in the permanent or viscous strain, ε _v, at unloading time of the tests, with increase in stress. Viscous strain thus provides a measure of the permanent change in the structure.

The analysis in Figure 1.7 clearly demonstrates that (i) the amount of delayed elastic strain, des, accumulated during the tests is not negligible and that (ii) it is not consumed within the mass and is recoverable at the end of the tests even well within the tertiary stages. The set of these types of creep and recovery curves illustrates the potentials of the SRRTs.

Figure 1.7 Stress–strain diagram. (a) Two values of elastic modulus, E, and strain trinity (elastic, delayed elastic, and viscous) for loading and unloading the 200 s SRRT of Figure 1.5 on Waspaloy at 1005 K and 650 MPa. (b) Stress–strain diagram, and the strain components for the longer‐term (2341 s) test in Figures 1.4 and 1.5 on Waspaloy at 732 °C (1005 K) and 650 MPa; the lower value of E (168 GPa) during unloading compared to 178 GPa during loading indicates structural damage during tertiary creep with slight increase in true stress.

Source: N. K. Sinha.

Stress–strain (σ–ε) diagrams for engineering materials are commonly “reserved” for constant displacement (nominally constant strain rate) tests from which some sort of “effective elastic modulus” is evaluated. Most textbooks on mechanics talk about this aspect and define engineering properties of materials accordingly. Hysteresis loop or stress–strain diagrams for constant stress (load) creep tests are unheard or not commonly discussed, perhaps because of uncertainty of the loading and unloading phases of the common creep tests using dead loads.

Computer‐controlled SRRTs provide unique opportunities for examining the effective elastic modulus during both loading and unloading times, and examine the differences (if any) associated with creep damages. An example is shown in Figure 1.7a for the 200 s SRRT of Figures 1.4 and 1.5 and this illustrates the possibilities of determining E during loading and unloading sequences. Such stress–strain diagrams for creep tests are unheard of to most materials scientists.

Comparing the E values of 180.0 GPa during the rise time to apply the full load and that of 177.6 GPa obtained for fall time (unloading) demonstrates that these values are complimentary and close to the dynamic values of previously undeformed and undamaged Young's modulus E of Waspaloy at comparable temperatures determined from seismic methods.

Figure 1.7b shows a stress–strain diagram for the longer (2341 s) duration SRRT of Figure 1.5. In this case, the E value of 178.2 GPa during the loading sequence is very close to those for short‐term loading and unloading periods in Figure 1.7a, but noticeably lower E value of 168.0 GPa obtained during the unloading period. This lower E value after longer‐term creep is indicative of a measure of the structurally “damaged” state of the material undergoing minimum creep rate and subsequent tertiary creep stage.

Figure 1.7 also reveals a particularly important aspect of constant‐stress creep tests by noting the experimentally determined amounts of the three components of strain: elastic, delayed elastic, and viscous. Most important is the fact that the delayed elastic contribution to the total strain is measurable and not negligible. In the case of the 200 s test, ε _d is comparable to ε _v, whereas it is significantly lower in the case of the longer‐term 2341 s test. There was, therefore, a significant contribution of delayed elastic (anelastic) strain to the total “inelastic strain” at the time of the minimum creep rate. Realistic rheological models for high‐temperature engineering applications cannot ignore the contributions of delayed elastic strain, known to be associated with grain‐boundary shearing processes (Sinha 1979).

Rheological models must be able to quantify the trinity or three‐component aspects of creep and relate to failure processes; this is covered in Chapters 5–8. It is shown in Chapter 5 that the delayed elastic effect could vary from a linear to a highly nonlinear response, but during primary creep the ratio, n _v/s, of stress exponents, n _v, for viscous flow (dislocation creep) and, s, for delayed elasticity could be very similar for crystalline materials in general. Since delayed elasticity has been linked strongly to grain‐boundary shearing processes, metallurgical and process engineering for superalloys may be directed toward (grain‐boundary engineering) increasing this n _v/s ratio to decrease the propensity for generating intergranular voids and cracks.

Figure 1.8 presents five sets of results for different specimens on the stress dependence of viscous strain rate during the primary creep obtained from short‐term SRRTs, and the corresponding minimum creep rate obtained from the long‐term tests. The same (or numerically very close) power law with a stress exponent of 11.0 applies for both n _v for viscous strain rate during primary creep and n _min for the minimum creep rate. These results and more are discussed in detail in Chapter 6 for a number of different materials in conjunction with development of rheological equations for the primary or transient stage of creep.