CREEP DEFORMATION MECHANISMS IN CARBIDE PRECIPITATE STRENGTHENED NICKEL-BASED SUPERALLOYS AT ELEVATED TEMPERATURES

Creep deformation of nickel-based superalloys at elevated temperatures is inherently dependent on the microstructural state of the material. Carbides have been observed to suppress intragranular and intergranular deformation rates at elevated temperatures by impeding dislocation motion within the grain and along the grain boundary. First, the rate controlling effects of intraand intergranular carbides as it relates to the grain boundary sliding are examined. A microstructurally sensitive, viscous description and model of grain boundary sliding is then presented. This work provides the concepts and mathematical formulation to model the rate controlling processes governing the grain boundary sliding associated with creep of carbide-strengthened superalloy Inconel 617 (IN617). The framework of the model considers both the microstructural state (size, volume fraction, and spacing of carbides) in the matrix, as well as along the grain boundary when determining the overall sliding rate. The model accounts for the role of carbides as they pertain to dislocation arrival/absorption into the grain boundary and the rate at which they glide and climb along the grain boundary plane, resulting in grain boundary sliding. It is considered that grain boundary sliding necessitates the supply of extrinsic dislocations from the matrix to facilitate sliding and, as such, the rate at which dislocations arrive to and are absorbed into the grain boundary dictates the overall sliding rate. Carbides along the grain boundary are then modeled as accumulation points for backstress which suppresses grain boundary sliding as a function of their size and spacing. At elevated temperatures, carbides within the matrix and along the grain boundary are subjected to diffusional processes resulting in time-dependent microstructure and mechanical response, requiring a detailed understanding of the rate controlling properties of both intraand intergranular carbides as they pertain to grain boundary sliding and creep deformation. Following this, a method of using stress relaxation tests carried out at 780 °C on IN617 specimens of various aging exposure times to examine the effect that the matrix microstructure exerts over the material’s deformation at elevated temperatures is explored. Included in this experimental work are quantitative microstructural assessments of IN617 specimens of various exposure times through the use of scanning electron microscopy (SEM). Once the microstructure of the material has been evaluated, the elevated temperature stress relaxation tests are utilized as a means of producing accelerated creep behavior via interconversion of stress relaxation data to creep strain data. At 780°C, above the solvus temperature of gamma prime (γ’), but below that of the chromium-rich M23C6 carbides, the microstructure could be regarded as constant. In doing this, the time dependent nature of the M23C6 carbide evolution was able to be eliminated, thereby allowing a “snapshot in time” with fixed carbide radius, volume fraction, and spacing values which could be quantified via SEM. As the inelastic sliding of the viscous grain boundary is asserted to provide the means of stress relaxation, holding constant the grain and grain boundary microstructure allowed for determination of the number of grain boundary dislocations (ngb) required to produce the corresponding amount of grain boundary displacement. This is achieved by analyzing the matrix and grain boundary dislocation behavior within the framework of a physics-based deformation model which couples the matrix dislocation release (nm) to the grain boundary dislocation population which – through prevailing glide and climb processes internal to the boundary that occur at the experimental stress and temperature – facilitate grain boundary sliding. A unification of the influence of the matrix and grain boundary microstructure on the creep behavior of IN617 is then provided, proffering a comprehensive and efficient tool for consideration in the design and analysis of carbide precipitate strengthened nickel-based superalloys for high temperature applications.

microstructurally sensitive, viscous description and model of grain boundary sliding is then presented. This work provides the concepts and mathematical formulation to model the rate controlling processes governing the grain boundary sliding associated with creep of carbide-strengthened superalloy Inconel 617 (IN617). The framework of the model considers both the microstructural state (size, volume fraction, and spacing of carbides) in the matrix, as well as along the grain boundary when determining the overall sliding rate. The model accounts for the role of carbides as they pertain to dislocation arrival/absorption into the grain boundary and the rate at which they glide and climb along the grain boundary plane, resulting in grain boundary sliding. It is considered that grain boundary sliding necessitates the supply of extrinsic dislocations from the matrix to facilitate sliding and, as such, the rate at which dislocations arrive to and are absorbed into the grain boundary dictates the overall sliding rate. Carbides along the grain boundary are then modeled as accumulation points for backstress which suppresses grain boundary sliding as a function of their size and spacing. At elevated temperatures, carbides within the matrix and along the grain boundary are subjected to diffusional processes resulting in time-dependent microstructure and mechanical response, requiring a detailed understanding of the rate controlling properties of both intraand intergranular carbides as they pertain to grain boundary sliding and creep deformation.
Following this, a method of using stress relaxation tests carried out at 780 °C on IN617 specimens of various aging exposure times to examine the effect that the matrix microstructure exerts over the material's deformation at elevated temperatures is explored. Included in this experimental work are quantitative microstructural assessments of IN617 specimens of various exposure times through the use of scanning electron microscopy (SEM). Once the microstructure of the material has been evaluated, the elevated temperature stress relaxation tests are utilized as a means of producing accelerated creep behavior via interconversion of stress relaxation data to creep strain data. At 780°C, above the solvus temperature of gamma prime (γ'), but below that of the chromium-rich M23C6 carbides, the microstructure could be regarded as constant. In doing this, the time dependent nature of the M23C6 carbide evolution was able to be eliminated, thereby allowing a "snapshot in time" with fixed carbide radius, volume fraction, and spacing values which could be quantified via SEM. As the inelastic sliding of the viscous grain boundary is asserted to provide the means of stress relaxation, holding constant the grain and grain boundary microstructure allowed for determination of the number of grain boundary dislocations (ngb) required to produce the corresponding amount of grain boundary displacement. This is achieved by analyzing the matrix and grain boundary dislocation behavior within the framework of a physics-based deformation model which couples the matrix dislocation release (nm) to the grain boundary dislocation population which -through prevailing glide and climb processes internal to the boundary that occur at the experimental stress and temperature -facilitate grain boundary sliding. A unification of the influence of the matrix and grain boundary microstructure on the creep behavior of IN617 is then provided, proffering a comprehensive and efficient tool for consideration in the design and analysis of carbide precipitate strengthened nickel-based superalloys for high temperature applications. To my mother, Linda -I am immensely indebted to you for being the person I have always been able to look up to and call my best friend. Everything I have accomplished I owe to you, your sacrifices, hard work, and resilience. You are the epitome of a role model and the proof that I turn to when I need to be reminded of how kindness, love, curiosity, and hard work allow us to make a lasting impact on the world. I could never repay you for all that you have given me.
vi To my sister, Cassie -thank you for always being a fortifying presence in my life and living it in a way which constantly reminds me that the pursuit of knowledge and science is so much more than what it appears on the surface.
To Sunny -for the tireless love and support you show in every way and for sharing in my enthusiasm when it comes to chasing dreams.
To Mike Dean -for being so much more to me and my family than I could have ever realized at the time. Your presence will always be missed, but your love always felt.   (1) represents the grain boundary/matrix interface and interface (2) represents the matrix with the external surface. (B) Stress is applied to interface (2), deforming the assembly from xo by dx. The initial loading elastically deforms the matrix. The elastic load results in a distance of x1 between interfaces (1) and (2), which remains constant throughout the application of the load. (C) Continued displacement of interfaces (1) and (2)        %) [2]. IN617 has found use in a myriad of components and applications within the aerospace industry, specifically in gas turbine design. In the basic design of the turbojet engine, the components which see the greatest temperature and stress lie in the combustor and turbine sections. As air flows into the turbojet engine, it is pressurized by the compressor and continues into the combustor, where fuel is injected and ignited. This combustion is then turned into mechanical work by the turbine, which in turn spins the compressor through use of a shaft. The hot gas then flows and expands through the turbine section, generating thrust. The temperature at which the gas leaving the combustor enters the turbine has a significant impact on engine efficiency; this parameter is referred to as the turbine inlet temperature. Generally speaking, the higher the turbine inlet temperature, the greater the associated enthalpic input into the turbine and subsequent efficiency.
The turbine inlet temperature has increased nearly 700 °C over the span of the last seven decades. With the demand for increasing temperatures, the material capability must follow suit.
IN617 is also a candidate material for the Generation IV next generation very high temperature nuclear reactor (VHTR) components. The VHTR design, unlike its light water (pressurized water or boiling water) reactor counterparts, utilizes the Brayton cycle in lieu of the Rankine cycle. In using the Brayton cycle, a thermal efficiencies in excess of 50% can be seen [3]. This increase in thermal efficiency is driven by greater process temperatures. Even with reactor core helium cooling, primary loop temperatures are expected to exceed the 1000 °C threshold.
The most important design considerations in these applications often Solute atoms affect dislocation mobility by affecting elastic and modulus interactions through distortion of the lattice and localized changes in shear modulus, respectively [5]. The effectiveness of a solute's ability to strengthen the matrix is proportional to the amount of misfit arising from differences in atomic radii. While substitutional solute atoms are only capable of affecting the motion of edge dislocations, interstitial solute atoms can impede both edge and screw dislocations due to the fact that they produce both dilatational and shear lattice distortion [6]. The strengthening of the grain boundary also presents another means of impeding dislocation motion. As dislocations from within the grain attempt to move (or transmit) from one grain to the next, the degree of misorientation between adjacent grains can make this process increasingly 4 difficult due to the discontinuity of slip planes [3], [6]. When sufficient temperature and stress are present, as dislocations enter and move through the more atomically disorganized (relative to the matrix) grain boundary, their motion produces grain boundary sliding. The more organized the grain boundary becomes, the less free volume exists, and the more difficult it becomes for dislocations to move. This is further compounded by the presence of grain boundary carbides, which act as barriers to dislocation glide. The precipitation hardening mechanisms occur in several different forms which somewhat parallel the means by which solid solutioning strengthens the matrix: stacking-fault strengthening, order strengthening, modulus misfit strengthening, coherency strengthening, and chemical strengthening. When a dislocation encounters a precipitate and is to continue its motion, it must either shear the precipitate, bypass it by way of Orowan looping, or climb over it. A single dislocation shearing a particle generates an anti-phase boundary, in which the stacking sequence of the particle's atomic arrangement is shifted. The energy required depends on the volume fraction, size, and spacing of particles, as these terms dictate whether a strong or weak dislocation coupling occurs. The strength or weakness of the coupling dictates the magnitude of the required, applied stress in order to move dislocations through the particles encountered. Once a critical precipitate radius is reached, Orowan looping becomes more favorable. Orowan looping occurs when a dislocation encounters a precipitate and is forced to bow due to the obstacle's presence. At the point when the maximum curvature of the dislocation is reached, the dislocation will meet around the backside of the precipitate and annihilate the portion of itself which has opposing signs, and will proceed to continue its motion. In its wake, it leaves behind a dislocation which has now formed a loop around the precipitate. This loop contributes to an increase in backstress for future dislocations which try to bypass the particle [4]. In nickel-based superalloys at temperatures at or exceeding 0.6-0.7Tm, when accompanied by moderate to high stresses, particle looping and shearing will give way to the more energetically favorable dislocation climb. A dislocation climb event requires the motion of a vacancy to the core of the dislocation such that it can move to an adjacent glide plane.
The significance of both matrix (intragranular) and grain boundary (intergranular) deformation in high temperature creep has been explored in depth by many authors since the 1950s. These studies have examined the effects of grain size, grain and grain boundary shape, various crystalline phase/structure formations, and the precipitation of various phases as a function of heat treatment, aging, and service exposure in many different engineering alloys. Grain boundary sliding as a primary deformation mechanism in Nickel-based superalloys is well understood and has been explored in depth over the last several decades. In

Justification of Thesis
The creep mechanisms of a crystalline material are generally described in terms of dislocation creep (low temperature, high stress) or diffusional creep (high temperature, low to moderate stress). One of the key differences in mechanisms between these two regimes can be explained by the increase in concentration of vacancies and the increase in thermal energy which promotes the ability for 7 vacancies to diffuse toward or away from dislocations. Without this increase in concentration and mobility of vacancies from sources to sinks (e.g. grain boundaries), edge dislocations are generally confined to glide within their respective slip system; however, as temperatures increase, dislocations become increasingly capable of climb. This becomes important when frictional stress due to dislocation-dislocation interaction approaches the order of magnitude of the applied stress or, most relevant to this thesis, when obstacles to dislocation glide are present, such as matrix carbides.
The crystal lattice has a relatively high degree of order and diffusive motion along the grain boundaries becomes the favored mechanism at lower temperatures.
This is due to the lower activation energy threshold required for diffusion in the grain boundary, as the misorientation characteristic of the grain boundaries of polycrystalline materials presents a larger free volume through which diffusion can occur. As temperatures exceed 0.6-0.7Tm, the vacancies within the matrix see a significant increase in their concentration and mobility due to the increase in thermal energy. When this increase in energy results in a total value that approaches that of activation energy of self-diffusion within the matrix, the increase in magnitude of intragranular and intergranular diffusive processes follow. Soula et al. [10] utilized micro-deposited ceramic gridding and microextensometry to investigate the qualitative relationship between the intragranular and intergranular deformation of γ'-strengthened NR6 material. It was shown that the amount of local deformation due to grain boundary sliding events is significantly greater than that which occurs due to matrix deformation (via activation of slip bands) and that the grain boundary sliding deformation was largest 9 in concentrated areas where slip band incidence was high. The authors observed that the increase in the macroscopic strain magnitude is proportional to the increase in both the number of grain boundary sliding events and slip band generations within the matrix. Prior to reaching the threshold strain of approximately 1%, the rate of increase of intragranular deformation was greater than that of the number of grain boundary sliding events, even though grain boundary sliding displacement dominated. As macroscopic strain increases beyond this strain threshold value, local deformation continued to occur at sites which have already undergone deformation (i.e. grain boundaries where sliding had already taken place and grains which had already experienced slip). Soula et al. [10] also noted that the localized deformation at the slip band -grain boundary junctions resulted in local deformation that was approximately twice that found elsewhere along the grain boundary under examination and four times that which was found elsewhere within the grain. These observations suggest that the ability for dislocations to move from within the matrix to the grain boundary play a significant role in the deformation mechanism and resulting strain rate. This is also seen in the work of Thibault et al. [11] on the same γ'-strengthened NR6 material. In varying the intragranular matrix microstructure by employing the use of different heat treatments, it has been shown that the ability of the dislocations to traverse intragranular obstacles during creep testing at 700 °C was directly reflected in the magnitude of grain boundary sliding strain which occurred.
Intragranular deformation during creep is associated with activation of slip bands, in which multiple slip systems/planes have become activated and create a proposed that the rate of dislocation absorption is fundamentally that of the rate of dislocation removal or annihilation within the grain boundary [12].
The removal of dislocations can occur via grain boundary vacancy diffusion-facilitated climb over grain boundary obstacles such as ledges, carbides, and/or triple points. It has also been proposed [12] that extrinsic grain boundary dislocations (EGBDs) can be annihilated by interaction with other EGBDs of the opposite sign moving along the same boundary and at triple points. As the structural order of the grain boundaries increase, yielding a reduction in their free volume, dislocation motion becomes more energetically demanding. In other words, as the number of coincident site lattice positions increases (i.e. the closer the grain boundary gets to the order of a perfect crystal), grain boundary free energy decreases, which then increases the activation energy required for grain boundary sliding.
It is widely accepted that grain boundary sliding is both an accommodation and deformation mechanism which produces measurable strain during creep. Grain boundary sliding models have been developed [13]- [16] to predict creep strain rate based upon the underlying assumption that grain boundary sliding is the dominant deformation process and is proportional to the measured creep strain. Wu and Koul [15] revised a model developed originally established by Langdon [13] to capture the significance that grain boundary microstructure -specifically the presence of M23C6 carbides -has on the creep behavior of nickel-based superalloys.
The significance presented by the models offered in References [14] and [15] is that both account for dislocation pile-ups that can occur when obstacles which prevent dislocation glide along the boundaries are encountered. These dislocation pile-ups give rise to backstress which counter the applied stress acting along the grain boundary. This backstress results in a lower effective stress acting along the grain boundary, and in turn, reduces the rate of the creep strain. This holds true so long as the carbides remain discrete [15].

Abstract
At elevated temperatures and moderate to high stresses, dislocation creep is considered the dominating mechanism of deformation, governed by the motion of dislocations through the matrix and along the grain boundary. Increased activity of matrix dislocations has been observed to accelerate grain boundary sliding at elevated temperatures which is considered here as a function of dislocation arrival rate to the grain boundary. Dislocations that arrive to and are absorbed into the boundary plane can contribute to grain boundary sliding as they glide under shear.
Therefore, the rate at which dislocations arrive to the boundary governs the boundaries ability to slide. Precipitation of secondary phases restricts dislocation glide, shifting the rate controlling mechanism from glide to climb. back stress accumulates at the interface, reducing the effective stress and sliding rate. The rate at which dislocations arrive to the grain boundary (a function of climb rate, particle size and spacing within the matrix), therefore, directly influences the sliding rate. Using the model described below, dislocation mobility through the matrix, as it relates to time-dependent microstructural changes, has been simulated and analyzed from which the rate controlling properties of matrix carbides is investigated.

Introduction
The operating environment of materials used in high temperature applications such as the aerospace and energy generation industries require that they exhibit strength, toughness, and creep resistance. As the demand for greater efficiencies has driven operating environments to higher temperatures and stresses, nickel-based superalloys have become one of the most prevalent families of materials due to their strength at high temperatures. The γ'-precipitate strengthened 14 nickel-based alloys has proven beneficial in many industrial applications of superalloys through the resulting increases in yield strength seen in low to moderate temperature environments. With increasing service temperatures, the γ' phase also confers creep resistance by acting as a barrier to dislocation motion forcing dislocations to loop, shear, and/or climb, thereby increasing the high temperature strength and prolonging the life of the material. However, at temperatures exceeding ~760°C, the benefits of the phase are lost as the γ' precipitate phase dissolves back into the matrix [17]. In applications at or above the γ' solvus, the most common precipitates found in many nickel-based superalloys are the metastable MC and M23C6 carbide formations.
Inconel 617 (IN617), a candidate material for high temperature applications exceeding 900°C, is a solid solution strengthened nickel-based super alloy owing its high temperature strength to misfit strains induced through the addition of elements such as Cr, Mn, Fe, and Co to the FCC Ni matrix. At these elevated temperatures, the solubility of these elements is exceeded within the solid solution resulting in segregation, precipitation, and coarsening of secondary phases in both intra-and intergranular regions. The elements Cr and C often favor the precipitation of carbides of the M23C6 type, which precipitate in a dispersed, discrete fashion in the early stages of creep (early on in the service life exposure). These carbides have been shown to dissolve and re-precipitate within the grain boundary as service time increases. In the grain boundary, these M23C6 carbides can form discrete networks with discernable spacing or continuous slabs which appear as "sheets" of carbides between grains [1], [14]. 15 The creep mechanisms of a crystalline material are generally described in terms of dislocation creep (low temperature, high stress) or diffusional creep (high temperature, low to moderate stress). This difference in mechanism can be explained by the ability for vacancies to diffuse. As the matrix has a relatively high degree of order, dislocation motion along the grain boundaries (Coble creep) becomes the favored mechanism at lower temperatures. This is due to the lower activation energy threshold required for diffusion in the grain boundary, as the misorientation of the grain boundaries presents a larger free volume through which diffusion can occur. As temperatures exceed 0.6-0.7Tm, the vacancies within the matrix increase in their mobility due to the increase in thermal energy; decreasing the energy barrier for self-diffusion within the matrix Experimental data pertaining to creep of IN617 and similar nickel-based superalloys suggests a synergistic relationship between matrix and grain boundary deformation. Kihara et al. [1] examined the creep behavior of IN617 subjected to various heat treatments. In the solution treated condition, carbides were dissolved into the solid solution such that intra-and intergranular regions were denuded of precipitates. This condition allowed for the precipitation of carbides within the matrix to occur during creep deformation. In the aged condition carbides in the matrix, which have already dissolved out in favor of grain boundary carbides, are not available to obstruct dislocation mobility. It was shown that fine, intragranular carbides, which precipitate rapidly during service exposure at 1000°C, suppress creep deformation. The creep data of this alloy showed that for specimens that were only solution treated, the primary (transient) creep rate was noticeably slower than that of the aged specimens and entered into steady state creep much earlier. The aged specimens exhibited a more classical sigmoidal behavior with a greater primary creep rate.
The work of Chomette et al. [9] on the same alloy mentioned above, in both aged and solution treated conditions, further reinforced this difference in primary creep performed at 30 MPa and 950 °C using specimens; the micrographs of the experimental samples taken over the 1 to 1000 hour aging process prior to testing closely resembled the behavior of the specimens tested by Kihara et al. Another key observation is that the steady state value of the crept specimens -regardless of the heat treatment procedure -proved to be approximately equal once controlled for the grain size.
Intragranular deformation during creep is associated with activation of slip bands, in which multiple slip systems have become activated and create a favorable system on which dislocation glide would occur. As slip bands are activated over the length of the grain, the primary inhibitors to dislocation glide are dislocation networks, secondary phases, or the grain boundaries. The energy of grain boundaries has been observed to be a factor in the dissociation of matrix dislocations and their subsequent absorption/transmission into and/or through the grain boundaries. These absorption delays resulting from the varying levels of grain boundary energies, affects dislocation mobility within the grain by generating pileups and subsequent backstress within the grain. It has been proposed that the rate of dislocation absorption is that of the rate of dislocation removal, or annihilation, within a characteristic grain boundary length [12]. This removal process can occur via grain boundary vacancy diffusion-facilitated climb over grain boundary obstacles such as ledges, carbides, and/or triple points. It has also been proposed [12] that extrinsic grain boundary dislocations (EGBDs) can be annihilated by interaction with other EGBDs of the opposite sign moving along the same boundary. As the structural order of the grain boundaries increase, this yields a reduction in free volume encapsulated within the grain boundaries whereby dislocation motion becomes more energetically demanding. In other words, as the number of coincident site lattice positions increases, grain boundary free energy decreases, which then increases the activation energy required for grain boundary sliding.
Carbides found in intragranular and intergranular locations have been observed to control dislocation mobility by shifting the rate controlling process of deformation from dislocation glide to dislocation climb. As dislocations encounter carbides, their ability to glide is halted. In the absence of Orowan looping or particle shearing (given the temperature and stress regime under consideration), for further dislocation glide to continue, dislocations must climb over the particle interface. A dislocation climb is the slower of the two processes (i.e. glide and climb), it is regarded as the rate-controlling deformation mechanism. As such, the size, volume fraction, coherency, and locality of these phases (matrix or grain boundary) have been observed to significantly alter the high temperature mechanical response of this material, particularly in relation to creep deformation facilitated via grain boundary sliding [14], [18]- [23].
The work examined here focuses on describing the concepts and mathematics of a microstructurally sensitive creep model which considers the interaction of matrix and grain boundary dislocations with carbides. This interaction is the basis of determining the overall creep strain in terms of viscous grain boundary sliding.

Model Overview
Several authors [1], [9], [21] have examined the creep behavior of nickelbased superalloys subjected to various solution treatment and aging processes which result in the presence or absence of intragranular carbides during creep testing. Kihara et al. [1] observed that aging at 1000 °C initially resulted in the precipitation of carbides in the matrix and along grain boundaries. With increasing aging time, however, carbides residing in the matrix eventually gave way to the more stable grain boundary carbides, denuding the matrix of precipitates. The precipitation of these meta-stable carbides within the grain had been considered as the reason for the suppression of strain rate during primary creep at high temperatures for IN617. A model developed by Dyson [19] and expanded upon by Manonukul et al. [20] aimed to directly calculate the significance that such intragranular precipitates had on creep strain. Experimental observations were supported by mathematical models which showed γ' precipitates within the matrix effectively slowed the rate of intragranular deformation. It was considered that, by forcing the matrix deformation mechanism to that of dislocation climb by way of presenting obstacles within the matrix, the more effectively dislocation motion was impeded. Further, the work of Liu and Jonas [24] on alloyed steels showed that the precipitation of titanium carbonitrides during stress relaxation testing was able to significantly slow the relaxation rate of the material.
Soula et al. [10] utilized micro-deposited ceramic gridding and microextensometry to examine the relationship between the intragranular and intergranular deformation of γ'-strengthened NR6 material. It was shown that the amount of local deformation due to grain boundary sliding events was significantly greater than that which occurs due to matrix deformation (via activation of slip bands) and that the grain boundary sliding deformation was largest in concentrated areas where slip-band/grain boundary incidence was high. The authors observed that the increase in the macroscopic strain magnitude is proportional to the increase in both the number of grain boundary sliding events and slip band generations within the matrix. Prior to reaching the threshold strain of approximately 1%, the rate of increase of intragranular deformation was greater than that of grain boundary sliding events, even though grain boundary sliding displacement dominated. As macroscopic strain increased beyond this strain threshold value, local deformation continued to occur at sites which had already undergone deformation -i.e., grain boundaries where sliding had already taken place and grains which had already experienced crystal slip. The authors have also noted that the localized deformation at the slip band/grain boundary junctions resulted in local deformation that was approximately twice that found elsewhere along the grain boundary under examination and four times that which was found elsewhere within the grain. These observations suggest that the ability for dislocations to move from within the matrix 20 to the grain boundary play a significant role in the deformation mechanism and resulting strain rate. The work of Fukutomi et al. [25] and Sheikh-Ali et al. [26] on Cd and Zn bicrystals, respectively, showed the grain boundary sliding rate increased five to ten times when accompanied by crystal slip versus that which occurred in its absence. This is seen in the work of Thibault et al. [11] on the same γ'-strengthened NR6 material. In varying the intragranular matrix microstructure by employing the use of different heat treatments, it was shown that the ability of the dislocations to traverse intragranular obstacles during creep testing at 700 °C was directly seen in the magnitude of grain boundary strain which occurred.
Furillo et al. [14] investigated the beneficial effects of grain boundary carbides on the creep response of a Nickel-based superalloy. Through conventional creep testing of specimens with and without grain boundary carbides, it was observed that the inclusion of the carbide phase was able to suppress or completely prevent grain boundary sliding. It was concluded that the presence of intergranular carbides provided points for dislocation pile up which generates a backstress resulting in a reduction in the sliding rate of the grain boundary. It was shown by examination of the creep behavior that the presence of grain boundary carbides resulted in an increase in the apparent stress exponent (from ~2 to ~15), which indicated a shift from grain boundary to matrix driven creep. Wu et al. [15],  ). As they approach the carbide, they begin to pile-up, as indicated by the dotted oval. As more dislocations enter the pile-up, the total number (np) over a given length generates a backstress which opposes the applied stress and reduces the rate at which trailing dislocations can glide to and enter the pile-up. Dislocations are then only able to escape the pile-up via climbing ( ) the distance equivalent to that of carbide radius .
Within the existing grain boundary sliding models [13]- [15], the rate-controlling variables in the calculated creep strain rate lies within the backstress term as a result of dislocation pile-up in front of grain boundary carbides and the rate of dislocation climb over them. These models indicate that the transition from primary to steady state creep is achieved when the backstress due to dislocation pile-up [ [27] have suggested that while it is possible that dislocations are generated within the grain boundaries, it is likely that the primary source of the increase in extrinsic grain boundary dislocations is due to absorption of lattice dislocations into the grain boundaries.
The concept of a "saturated" grain boundary due to matrix dislocation absorption is reinforced by the work of Pshenichnyuk et al. [28]. This population is a function of the lattice dislocations which are absorbed by the grain boundary and rendered as glissile grain boundary dislocations.
Matrix dislocations, upon entrance into the grain boundaries, will either be transmitted through or dissociate within the grain boundaries [6], [11]. As the slip band spacing generated during creep is presumed to be significantly larger than that which is produced in fatigue applications, the lack of adjacent grain slip system symmetry makes transmission difficult [7], and therefore is not considered in the model presented herein. If a dislocation is absorbed and not instantly locked (i.e. the dissociation produces a glissile component), there are two different dislocationdislocation interactions which will ensue [26], [29]. In the case of dislocations of the opposite sign, they result in micro-incompatibility which produces localized grain boundary sliding, that could then resolve itself, resulting in no measurable strain.
On the other hand, dislocations of the same sign could produce incompatibility resulting in a measurable grain boundary sliding. The entry points and density of dislocations introduced into the grain boundary is a factor of adjacent grain orientation (slip band alignment and spacing) and grain boundary energy (typically 24 characterized by the CSL boundary values) [6], [13]. The number of glissile grain boundary dislocations contained within the boundary has been described as being inversely proportional to the boundary viscosity. This viscosity, , and its relationship to dislocation density, was first proposed by Ashby [30] and takes the form of: where k is Boltzmann's constant, T is the exposure temperature (Kelvin), b is the Burgers vector, b D is the grain boundary diffusion coefficient, is the linear density of extrinsic dislocations within the grain boundary, and is the grain boundary length.
As the grain boundary viscosity is inversely proportional to the extrinsic grain boundary dislocation density [30] for a given sliding rate, the dislocations must travel faster within a smaller population and therefore the energy required for their motion increases as a function of their velocity squared. Therefore, the sliding rate of the grain boundary can be considered a time dependent parameter based upon the rate at which matrix dislocations are absorbed into the grain boundary and the rate at which the glissile components can slide. As such, rate controlling obstacles (i.e. carbides) in the matrix must also been taken into account when considering the overall grain boundary sliding rate.
As described above, grain boundary sliding and the rate at which it occurs is dependent on dislocation mobility within the grain and along the boundary. Of 25 particular interest in the current work is how the size and volume fraction of intraand intergranular carbides affect the overall grain boundary sliding rate. To accomplish this, a microstructurally sensitive, viscous grain boundary sliding model has been developed which considers the rate of dislocation glide and pile up along the boundary as a function of the supply rate of dislocations from the matrix.
This model is described in the following section.
The cohesive zone relies on the elastic traction-displacement law applied at the matrix and grain boundary interface, and is described by [4]: where is the tangential traction gb K is the boundary stiffness, and el gb u is the elastic displacement of the grain boundary. The elastic displacement is defined as: where app T is the applied traction (generated by the applied load) and back T is the opposing force due to the accumulation of backstress. This produces a net traction, net T .

Grain Boundary Backstress
From Wu et. al [15], the backstress exerted on a moving dislocation a distance x from the particle is given by: where is the number of dislocations in the pile-up, ν is Poisson's ratio, is the shear modulus of the grain boundary, and b is the Burgers vector. For a dislocation to glide over the total distance between two carbides, the applied work ( ) must be greater than the resistance due to backstress. This is calculated through the To consider the number of dislocations, gb n , the density, gb As mentioned previously, the number of dislocations in the grain boundary, gb n , is considered here to be equal to the number of dislocations which have arrived to and been absorbed by the grain boundary, m n .

Matrix Deformation and Dislocation Availability (nm)
Within a deforming matrix, consider the two populations of dislocationsmobile (gliding) and immobile (pinned). The pinning refers to the fraction of the total population which will be rendered immobile by the carbides at any given time.
The rate of change of the gliding dislocation density within the matrix, g   , is expressed as the sum of the rate of dislocations which are able to successfully climb over the carbides pinning them plus the rate of dislocation generation via multiplication minus the rate of mobile dislocations which get trapped at carbides. This is written as: where c  is the instantaneous density of pinned dislocations, c x is the probability that a pinned dislocation will escape via climb, c r  is the climb rate, g  is the instantaneous density of mobile dislocations, g x is the probability that a mobile dislocation will become pinned by encountering a carbide (taken as unity), ̇ is the dislocation pinning rate, and ̇ is the generation rate of mobile dislocations.
Similar to the work of Dyson [19], it is assumed that the population of mobile gliding dislocations saturates rapidly due to the presence of matrix carbides, thereby driving the left hand term of Eq. [2.4-16], ̇ , rapidly to zero. Additionally, the dislocation generation term, g Q  , is assumed to be orders of magnitude smaller than the remaining terms in Eq. [2.4-16] and is neglected [33]. The two assumptions above simplify Eq. [2.4-16] to: c c g g c g x r x r The probability that a pinned dislocation is able to climb over a particle, c x , is considered proportional to the product of the dislocation line length in contact with the particle (approximated as the volume fraction, p  ) and Burgers vector, b , and inversely proportional to the particle size, p r . Therefore, c x , can be approximated as [33]: The rate of dislocation escape via climb over precipitates, c r  , is the ratio of dislocation climb velocity, c v , to Burgers vector, b : The rate of the dislocation pinning, ̇ , is the ratio of the dislocation glide velocity, where is the diffusion coefficient for self-diffusion, is the average shear stress acting within the matrix, k is Boltzmann's constant, and T is temperature (Kelvin).
It is with this set of equations that the rate of matrix strain, as well as the matrix regarded as time-independent. It is also important to realize that the rate of pile-up growth and backstress generation will ultimately be affected by the characteristic length over which sliding occurs (e.g. grain boundary carbide spacing, grain boundary length, etc.) and grain boundary viscosity. As Ashby [30] identifies, this is due to the fact that the grain boundary has a saturation limit of ( / ).

Intragranular Dislocation Mobility
The    The number of dislocations released for Conditions 1-3 and Conditions 1a-1c are presented in Figure 2.5-5 and Figure 2.5-6, respectively, as the ratio of dislocations released from the matrix with respect to the increase of mobile dislocations within the matrix.  To characterize the effect of the inextricably related matrix carbide radius ( p r ) and volume fraction ( p  ), the spacing ( p  ) was calculated using Eq.  The matrix carbide spacing was calculated as a function of time and plotted alongside the number of matrix dislocations which are able to be absorbed by the grain boundaries at a stress of 24.5 MPa and 1000 °C. Conditions 1a, 1b, and 1c were chosen to be plotted against spacing values as the radius growth rates and volume fraction rates were maintained constant in these conditions, which is assumed to be reflective of the identical microstructure evolution of three specimens of the same material exposed to the same temperature for different aging durations.
It can be seen in Figure 2 al. [18] where an increase in carbide spacing resulted in a greater steady state creep rate. This is realized upon examination of Figure 2.5-7, whereby increases in the carbide spacing term are accompanied by an increase in the steady state matrix dislocation release rate. It is considered that grain boundary sliding necessitates a supply of extrinsic dislocations that are released from the matrix during creep. As carbides act as obstacles to dislocation motion, their presence in the matrix suppresses the rate at which dislocations can arrive to the boundary to facilitate grain boundary sliding.

Conclusions
Carbides along the grain boundary act as obstacles and provide points of dislocation accumulation resulting the generation of back stress. The rate of accumulation is therefore considered the mechanism in which steady state is achieved. Stress relaxation tests were used to identify model parameters, as discussed in Reference [35], to simulate dislocation mobility within the grain and along the grain boundary.

Introduction
The relationship between intragranular slip and grain boundary sliding were first discussed in the works of Langdon [13], McLean and Lin [27], Ishida and Henderson [36], and Hirth [7]. It was observed that the population of grain boundary dislocations increased significantly with strain [36] which was attributed to the absorption of extrinsic matrix dislocations into the grain boundary [12], [27]. The magnitude of grain boundary sliding was observed to increase with increasing slip activity in the matrix [10]. It has been suggested that grain boundary sliding, facilitated by dislocation glide, necessitates the absorption of extrinsic matrix dislocations, which subsequently glide along the boundary plane [12], [26], [28], [34]. The rate of dislocation arrival to and along the grain boundary is suppressed through the precipitation of carbides which act as obstacles, changing the rate controlling mechanism of deformation from dislocation glide to dislocation climb [13], [14], [18], [19].
In Part I of this paper [35], the rate at which dislocations arrive to the grain boundary from the matrix, m n  , has been described as: where m  is the shear strain acting on the grain, p r is the matrix carbide radius value,  is a material constant,  is the mobile dislocation density within the grain, where µ is a material constant (0 < µ < 1), gb n is the number of mobile grain where v is Poisson's ratio, gb E is the grain boundary modulus,  is the grain boundary viscosity, and s u  is the grain boundary sliding rate. Under the assumption that the number of mobile extrinsic grain boundary dislocations, gb n , is equal to the number that have been released from the matrix, m n , Eq. [3.  can be rewritten as: such that the contribution of matrix dislocations to the grain boundary sliding rate is defined.
In the following work, a series of aging and stress relaxation tests have been conducted as a means of parameter determination for the model described in the previous work [35]. Aging at 1000°C for various exposure times were used to produce unique values of carbide size, spacing, and volume fraction. Following aging, stress relaxation testing at 780°C was conducted under compressive load, with an initial stress of 200 MPa. Microstructural analysis of carbides (aging), through SEM imaging, and associated mechanical response (stress relaxation) provide the necessary parameters for analyzing the rate controlling properties carbides on the grain boundary sliding rate as described above.

Material
The material of interest in the current study is Inconel 617, a solutionstrengthened nickel-based superalloy and a candidate material for the next generation of nuclear reactors. At temperatures in the range of 649-1093°C, the major strengthening phases have been determined to be the M23C6 type carbide which have been shown to precipitate early in the aging process [1] residing preferentially along grain boundaries, followed by intragranular regions [2], [17], [37], [38]. Standard composition can be seen in Table 3.3-1. Material used in this study was hot rolled and obtained in the solution treated condition. All specimens were subjected to a two-hour solution heat treatment at 1200°C followed by a water quench to dissolve any remaining secondary phases.
Intragranular MC carbides can still be observed after solutioning and are oriented in bands along the rolling direction, as shown in Figure 3  Microstructure (50X magnification) after solutioning at 1200 °C for 2 hours followed by a water quench. Intragranular carbides can still be observed in bands that run along the rolling direction (horizontally with respect to the micrograph). All samples were subjected to the solution treatment to ensure the same initial microstructure prior to any further aging treatments and testing.

Specimen Aging
The heat treatments for the four specimens are detailed in Table 3.3-2. As the in-service temperature range of interest is 950 °C -1000 °C, an aging temperature of 1000 °C was chosen in order to best replicate the microstructural evolution which would be expected for IN617 during its service life. Solution and aging treatments of the as-received IN617 specimens were conducted in furnace with a digital temperature readout from a thermocouple placed within the furnace tube. Ice water bath quenching was performed immediately after both the solution and aging treatments. Surface preparation was conducted using a progression of coarse metallurgical grinding paper (600 grit) to diamond polishing slurry (0.5 micron). Electro-chemical etching was performed using Carpenter's Reagent Etchant (8.5 grams FeCl3 / 2.4 grams CuCl2/122 ml alcohol/122 ml HCl/6 ml HNO3) and a 12V power supply with a 0.225 -0.250 Amp current output. After etching, the specimens were analyzed using a Jeol JSM-840 SEM with a tungsten filament supplied with a 15-18kV accelerating voltage. In addition to those listed in Table 3.3-2, an additional specimen (OB-1) was solutioned in an identical fashion to the stress relaxation specimens and aged for at 1000 °C for seven hours.
This was done as a point of comparison in order to verify carbide measurements made for the SR-1, SR-2, and SR-3 test specimens; these additional observations are presented in Table 3.3-2.

Stress Relaxation
Stress relaxation tests were carried out following ASTM E328 as a guideline for specimen dimensioning and experimental procedure. The stress relaxation specimen dimensions and test parameters are listed in Table 3.3-3. Testing was conducted on a servo-hydraulic mechanical testing system (MTS) equipped with a high temperature furnace. The test temperature was recorded for the entire test duration through the use of two thermocouples welded to adjacent sides of each test specimen. Each thermocouple was connected to a TestStar II control system which monitored and maintained the desired temperature. The stress relaxation procedure is illustrated in Figure 3.3-3.  (2), the stress applied to each test specimen was maintained while the furnace was brought up to temperature and thermal expansion occurred. The Ramp segment (3) involved load ramping to approximately 17 kN to achieve the desired initial compressive stress of 200 MPa. The Hold/Stress Relaxation segment (4) was executed in displacement control in which each specimen was permitted to relax. When sufficient time had passed such that an identifiable saturation in stress was observed, the furnace was turned off and the specimen and test equipment were allowed to cool (5) while maintaining a constant applied stress. Once cooled to room temperature, the specimen was unloaded (6) and removed from the testing machine.
The stress relaxation test was initialized by applying a small compressive load (~1 MPa) to hold the test specimen in place between the grips. During heating to 780°C at 1 MPa, the specimen displacement was monitored in order to ensure the thermal expansion of it and the test equipment had stabilized prior to loading.
A test temperature of 780 °C was chosen, as to fall above the γ' solvus temperature of approximately 760 °C [17] to prevent unintentional γ' precipitation, as well as to suppress M23C6 carbide evolution [38]. Once the test temperature was achieved and a negligible fluctuation in displacement due to thermal expansion was observed, an initial stress of 200 MPa was applied at a rate of 1.4 kN/s (strain rate of ~1*10 -4 s -1 ). Based on the strain-rate sensitivity of IN617 presented by Rahman et al. [39], the target strain rate was chosen such that test specimens remained with the elastic regime during the load ramp process, as the proportional limit has been shown to vary with strain rate. Upon achieving the initial stress, the control mode was shifted from force to displacement control, maintaining constant displacement through the stress relaxation test period.

Aging
Microstructure post aging for 3, 7, and 30 hours can be seen in Figure 3 Table 3.4-1. Grain size was determined from the linear intercept method and was also observed to increase with increasing exposure time.

High Temperature Stress Relaxation
In order to verify that the loading remained in the elastic regime of the material, the linearity of the load-displacement curves for the SR-1, SR-2, and SR-3 was verified, as shown in Figure 3.4-5 and detailed in Table 3.4-2.  Stress relaxation testing results can be seen in Figure 3.4-6(A) for the three aging conditions (0, 3, and 30 hours), from which the role of aging on the relaxation response can be readily observed. The magnitude and rate of relaxation is significantly reduced with thermal exposure prior, however, little variation in the relaxation response is observed for aging between 3 and 30 hours. The total strain of a material undergoing stress relaxation can be decomposed into its constituent parts as follows: where , , and are the elastic, inelastic, and plastic strain components. As where  is the relaxation rate. The inelastic strain rate as a function of stress is presented in Figure 3.4-6(B).  In the 0.6-0.7Tm temperature regime at moderate to high stresses, the presence of the M23C6 matrix carbides results in dislocation climb being the ratecontrolling mechanism. Liu and Jonas [24] demonstrated the significance of matrix titanium carbonitrides in alloyed steel through stress relaxation testing at various initial stresses in the temperature range of 850 -1050 °C. El-Magd et al. [21], Balantic et al. [18], and Mukherjee [31] showed either experimentally or through analysis of previously published work how the presence of matrix precipitates interrupts dislocation glide, slowing down the rate of deformation by forcing dislocations to climb. Mukherjee [31] identified dislocation climb as being the primary deformation mechanism when the stress exponent n is approximately equal to a value of 4 or 5. Furillo et al. [14], through conventional creep testing of specimens with and without grain boundary carbides, concluded that grain boundary carbides were able to suppress or completely prevent grain boundary sliding. A change in apparent stress exponent from ~2 to ~15, with the inclusion of grain boundary carbides, represented a shift from grain boundary to matrix driven creep. Determination of the stress exponent from the stress relaxation data presented in Figure 3.4-6 was calculated through the Zener-Holloman relationship: where Z is the Zener-Holloman parameter,  is the strain rate, B, is a material specific parameter, E is the temperature dependent elastic modulus, Q is the 66 activation energy for creep, n is the stress exponent, T is temperature in Kelvin, and R is the universal gas constant. Rearranging Eq. [3.4-3] and solving for n yields: from which the stress exponent can be plotted as a function of time and stress, as shown in Figure 3   The stress exponent, for all aging conditions, initially exhibited values of n > 15 suggesting the initial relaxation behavior was governed by matrix deformation; facilitated by through motion of matrix dislocations [14]. It is expected that the supply of dislocations to the grain boundary occurs rapidly in the initial stages decreasing in rate for values of n > 3, which represents a transition from combined matrix/grain boundary deformation to grain boundary dominated deformation. As such, it would be expected that matrix carbides govern the initial relaxation behavior, n > 3 while grain boundary carbides govern the later stages, n < 3.
As creep is a constant load process, ascertaining the creep behavior from stress relaxation behavior of a material is not a direct process. Determining a material's creep behavior from stress relaxation data (or vice versa) is often implemented when limited data is available, as relaxation modulus and creep compliance are the most direct way to characterize and predict a material's behavior as a function of time, stress, and temperature. This has been demonstrated with success using various numerical and analytical techniques [40]- [43].

Model Simulation
A analytical technique originally presented by Vorotnikov and Rovinskii [42], and further explained by Jung et al. [44], provides a straight forward method for converting stress relaxation data into creep strain data given for an equivalent, constant creep stress value. Vorotnikov and Rovinskii [42] presented a relationship which was used to successfully relate stress relaxation data to previously published creep data. With an initial elastic strain, ε0, the rate at which the conversion of elastic strain to plastic strain via viscoelastic deformation has been characterized as: Jung et al. [44] is of the form: Once values for k and p have been determined, Eq. [3.5-2] can be used in conjunction with the stress relaxation data to generate creep strain data given a constant creep stress ( creep  ) as: where M is the Taylor Factor. The rate of grain boundary sliding, ̇ , is determined by Eq. [3.5-7] over the interval [ , ]. The net traction force required over the specified time interval is then calculated as: The rate equation presented in Eq. [3.2-2] can be rearranged as shown in Eq.
[3. [5][6][7][8][9][10][11] in order to relate the number of dislocations that are released from the matrix, , to that which is required within the grain boundary characteristic length, , to facilitate the grain boundary sliding process:   [3.5-11] This rearrangement, along with Eq. [2.4-33] presented in the previous work [35] links the matrix microstructure and dislocation kinetics to that of the grain boundary. In order to gain an understanding of the behavior of the dislocation kinetics within the matrix and grain boundary, the uncoupled version of the grain boundary sliding model is applied to the test data using Eqs. , and recovery rate over a given grain boundary characteristic length.
The high temperature stress relaxation test data for specimens SR-1, SR-2, and SR-3, through the previously described conversion process, are used to produce equivalent creep strain curves, as shown in Figure 3.5-1(A). As grain boundary sliding is asserted to be the expression of the stress relaxation mechanism, the inelastic grain boundary displacement can be determined through application of Eq.
[3.5-9] to the stress relaxation data. The calculated inelastic sliding displacement for each stress relaxation data set is shown in Figure 3.5-2.      The number of dislocations within the grain boundary, , as a function of time for each specimen is calculate, normalized, and plotted against time in Figure 3.5-7.
The normalization constant,  , is defined as: [3.5-12] where gb L is the grain boundary length, b is the Burgers vector, and is a material constant. Eq.  The previous work [35] used the matrix deformation model to study several different matrix carbide ripening (growth, coarsening, dissolution) time profiles in order to assess the effects on the number and rate of matrix dislocation release as a function of matrix microstructure. In a similar fashion, the matrix deformation model is used to analyze the matrix dislocation release behavior of test specimens SR-1, SR-2, and SR-3. The difference now being that, unlike the matrix microstructure studies performed in the previous paper [35], the microstructure of the three test specimens is constant over the time domain of 80 interest, as detailed in Table 3.4-1. The matrix dislocation release as a function of time for each of the stress relaxation test specimens is shown in Figure 3.5-8. The initial simulation behavior of exhibited instability within the first ten time steps, which is believed to be a result of the assumed initial conditions that were assigned. Beyond this region, the number and rate of dislocation release from the matrix ( and ̇ , respectively) can be compared with the number and rate of dislocation absorption within the grain boundary ( and ̇ , respectively) that are required to facilitate the grain boundary sliding that acts as the stress relaxation mechanism in IN617 test specimens. In comparing Figure 3.5-7 and Figure 3.5-8, it can be seen that the magnitude of is not always larger than that of which requires further explanation. With respect to the physics-based model used to generate the results, shown in Figure 3.5-7 and Figure 3.5-8, the grain boundary sliding process demands the value of in order to produce the inelastic strain which ultimately provides the relaxation of stress. The inherent assertion of the model then is that the matrix is, at a minimum, capable of supplying exactly the value of . In considering the values calculated for the mobile dislocation density within the grain for each test specimen as was done in the previous work [35] and the work of Alexandreanu et al. [12] which suggests that the grain is capable of producing more dislocations than the grain boundaries are capable of accommodating, it is presumed that such differences in magnitude seen between and in Figure 3     It is expected, from inspection of Figure 3.5-12, that the microstructure has remained constant during testing of SR-2 and SR-2-R1, as the stress exponent for a given stress was consistent between both cycles. The difference in relaxation response of SR-2 and SR-2-R1 occurs predominately within the first 36 hours of relaxation, after which both cycles exhibit similar relaxation rates, as shown in  With the appropriate initial conditions, it was desired to execute the coupled, physics-based model in a serial fashion depicted in Figure 3.5-15. As grain boundary sliding had already occurred in the SR-2 test specimen, the initial conditions used for the SR-2-R1 simulation changed relative to that of the SR-2 specimen; inelastic displacement accumulated from previous cycle, the elastic strain of the grain along the grain-grain boundary interface, as well as the number of dislocations within the grain boundary. Beyond 22 hours of stress relaxation, the calculated values of , , and for all specimens began to exhibit saturation behavior, as shown in Figure 3.5-6 - Figure 3.5-8. The calculated inelastic grain boundary sliding displacement for all specimens, as shown in Figure 3.5-2, did not exhibit the same saturation, but rather appeared to trend toward a linear behavior.
In considering the stress exponent value and observation of these simulation behaviors, it was determined that 36 hours (denoted by B t in Figure 3.5-14) would be the point in time from which to restart the SR-2 simulation in an attempt to predict the SR-2-R1 test results. After 36 hours, the stress exponent of the SR-2 specimen fell below a value of three, which evidenced a shift in mechanism away dislocation climb-controlled creep [14]. Using the same methodology by which the simulation results were produced in the previous section, the profiles for and were extended by re-running the SR-2 simulation to the 36 hour threshold.
Analysis of the simulation results showed that at 36 hours, the values of and for the SR-2 specimen had saturated to a reasonable extent.  Overview of the initial (SR-2) and repeated (SR-2-R1) stress relaxation tests accompanied by the process flow for the simulation used to predict the creep strain behavior of SR-2-R1. The initial stress relaxation test consists of loading the specimen (SR-2) to a compressive stress depicted by Point (1). The specimen relaxes to Point (2) and is unloaded along the path made by Point (2) and Point (3). The reloading to perform the second stress relaxation test of the specimen (SR-2-R1) follows the path from Point (3) to Point (4), which is the same stress magnitude as the initial stress achieved at Point (1) from the previous test. The specimen then relaxes to a stress state represented by Point (5), and is then unload from Point (5) to Point (6). The microstructural measurements taken from the pre-test SR-2 specimen are held constant as the test temperature is established such that no microstructural change will occur. The value of ngb at 36 hours from the simulation results presented Figure 3.5-7(B) is used as the initial condition for the grain boundary dislocation population for the SR-2-R1 simulation. The initial value of nm for the SR-2-R1 simulation is taken to be zero, along with the initial value of the dislocation pileup, np. The initial strain, ε0, is the irrecoverable strain of the matrix which occurred during the first stress relaxation experiment (SR-2).
The initial value of for the SR-2-R1 simulation is taken to be zero. This is due to the fact that the load was removed between the initial (SR-2) and reload (SR-2-R1) stress relaxation tests. In the absence of an applied force, there is no motivation for the dislocation pile-up to form. These initial conditions can then be applied to Eq. [3.2-2]. The elastic strain which was generated at the grain-grain boundary interface from the sliding during the SR-2 stress relaxation test results in 88 a residual elastic strain of the matrix. As the grain and grain boundary remain coherent, the amount of elastic strain which has occurred within the grain -and subsequently accommodated by the viscoplastic sliding of the grain boundarycannot be recovered. This irrecoverable strain can be simply envisioned as illustrated in Figure 3.5-16. MPa at 780 °C. Figure 3.5-17 shows the curve-fitted creep strain behavior (converted from SR-2-R1 stress relaxation data via the previously presented [35]) compared to that of SR-2-R1 model simulation prediction using initial conditions established from the first stress relaxation test (SR-2) as described above.  The model somewhat overshoots the saturation value of the primary creep regime the model and predicts a slightly higher value than what is seen experimentally.
The rate and value of creep strain saturation were found to be sensitive to the constant value selected to represent the grain boundary viscosity, η. Sensitivity studies performed in the generation of Figure 3.5-3 - Figure 3.5-6 also revealed that depending on the value of η chosen, the magnitude to, and rate at which, backstress accumulates is directly impacted.

Discussion
The value of the grain boundary viscosity influences the grain boundary sliding rate through the relationship established in Eq. [3.5-9]. In each of the model simulations for the test specimens, a simplifying assumption was made in which the grain boundary viscosity was assumed to be constant with respect to time. This is similar to the approach taken by Raj and Ashby [45] in their treatment of grain boundary viscosity with respect to grain boundary diffusion facilitated sliding, which they approximated through the use of Eq. [3.6-1]: 3 1 132 where is the grain diameter, is Boltzmann's constant, is temperature, is the diffusion coefficient for grain boundary diffusion, is the grain boundary thickness, and is the atomic volume. Ashby [30] where is the density of grain boundary dislocations and is a given unit length. As dislocations enter the grain boundary from within the matrix, the dislocation-to-dislocation spacing decreases. As Ashby indicated, the power required for dislocation glide varies in a manner proportional with the square of its glide velocity. More directly, as approaches the maximum value of (1/λ), spacing decreases, followed by a decrease in boundary viscosity [30]. As such, an improvement on the physics-based deformation model presented in the previous work [35] is likely to come from the incorporation of a time-dependent viscosity term in the form of: Analysis of the simulation results, specifically the rate at which dislocations enter the grain boundary from the matrix and enter the grain boundary pileup, provides a basis that suggests the grain boundary viscosity may be best described through use of a power law relationship, initially at a maximum value which subsequently saturates to a minimum. This can be physically rationalized by realizing that prior to subjecting the material to a creep (or stress relaxation) service environment, the grain boundary dislocation spacing can be taken to relatively large as the grain has not been releasing dislocations for grain boundary absorption. As time progresses, more dislocations enter and occupy the grain boundary volume, thereby decreasing the dislocation-to-dislocation spacing.
The grain boundary viscosity will decrease until it reaches its saturation limit which corresponds to the grain boundary's inability to absorb more dislocations until climb or annihilation occurs [12], [28].
A comparison of Figure 3.5-8 with the stress relaxation data suggests that even though the difference in matrix microstructure of test specimens SR-2 and SR-3 results in a rather significant difference in the number of matrix dislocations released, the grain boundary carbide spacing exerts influence over the amount of relaxation. As the difference in grain boundary carbide spacing between test specimens SR-2 (1.39 microns) and SR-3 (1.16 microns) is marginal, it is then reasoned that the difference in the number of dislocations which can be accommodated is proportionally marginal. This is supported by the fact that the dislocation pileup within the grain boundary saturates to a lower value and at a faster rate than calculated for the SR-1 specimen, as shown in Figure 3.5-6. By comparison, the back stress calculated for SR-2 and SR-3 reaches its maximum value quicker, thereby reducing the amount of effective stress which is promoting the grain boundary sliding (and subsequent relaxation) in a more rapid fashion than which occurs in test specimen SR-1. With respect to the grain boundary, as seen in Figure 3.5-3 the backstress appears to saturate to a similar value for all stress relaxation specimens, whereas the net stress shown in Figure 3.5-4 appears to saturate to different values unique to each specimen. This arises as a result of the climb-facilitated recovery rate being different for each specimen due to differences in grain boundary carbide radius.
The solutioned-only test specimen, SR-1 -which had negligible M23C6 matrix and grain boundary carbide precipitation -showed drastically different model simulation results. Figure 3.5-8 reveals that the number of matrix dislocations released, as well as the rate, is significantly higher than that of either SR-2 or SR-3 specimens. With the increase in matrix dislocation release stemming from a clean matrix, the number of dislocations able to be absorbed by the boundary with a greater characteristic length in which to accommodate them, the amount of grain boundary sliding and inelastic strain increases significantly, as seen in Figure   3.5-2.
For specimen SR-1, the characteristic length of the grain boundary, , was approximated as the length of one side of a hexagon with the simplified circular grain (measured diameter, ) circumscribed within it. As such, the length over which the dislocations are able to pile up was significantly larger than the other specimens which contained grain boundary carbides. The simulation results shown in Figure 3.5-5 indicate a noticeably higher value of for specimen SR-1 relative to that of SR-2 and SR-3. This observation is not initially intuitive, as SR-2 and SR-3, have shorter characteristic lengths over which the dislocation pile-ups are forming compared to that of SR-1. This may suggest, in agreement with the work of Wu et al. [15], that the grain boundary carbide spacing exerts, in addition to the matrix microstructure, a significant influence over the amount of grain boundary sliding which can be expressed.

Conclusions
Four  [8], [17]. In doing this, any changes in precipitation beyond that which occurred in the prescribed heat treatment were effectively eliminated, allowing for experimental data to be produced with a constant microstructure that could then be used as input for the previously established physics-based model [35]. The initial stress was chosen such that the test conditions would provide the requisite combination of temperature and stress such that high temperature dislocation creep would be the active mechanism [46], but low enough such that the material yield strength at the test temperature would not be exceeded.
The stress relaxation rate observed for test specimens SR-1, SR-2, and SR-3 was then used to produce equivalent creep curves using a graphical interconversion technique [42]. Using the methodology established in the previous work [35], the physics-based model could be used to examine the physical premise coupling the intragranular and intergranular regions through the dislocation release and subsequent grain boundary dislocation population terms, and .
Unlike in the preliminary simulations, the simulations used to analyze the SR-1, SR-2, and SR-3 specimens considered the as-tested conditions, in which the matrix and grain boundary carbide size and spacing were constant. The simulation results confirmed the preliminary assessments, showing that with increasing matrix carbide size and volume fraction (accompanied by the corresponding decrease in spacing) resulted in fewer dislocations being able to arrive to the grain boundary. Given the difference in values of for the SR-2 and SR-3 specimens being noticeably larger than that which is seen in their corresponding values (shown in Figure 3.5-8 and Figure 3.5-7, respectively), this suggests that grain boundary carbide spacing plays a significant role in the formation of the pileup and must be considered.
An additional stress relaxation test was performed on the SR-2 test specimen (SR-2-R1), in which the same initial stress of 200 MPa was reapplied after the initial relaxation exhibited saturation behavior. Initial conditions for terms , , , and elastic strain within the matrix were determined for the SR-2-R1 test specimen through running of the uncoupled constituent matrix and grain boundary sliding deformation models for test specimen SR-2. The initial As the rate at which dislocations enter the grain boundary from the matrix, the characteristic length of the grain boundary, as well as the size and density of M23C6 grain boundary carbides, is almost always a unique combination of parameters which dictates the grain boundary dislocation population and kinetics. Therefore, a viscosity function that is microstructure-independent would likely be most advantageous. sliding-facilitated creep crack growth [29], [47], [48]. The calculation and incorporation of a critical displacement of sliding within the grain boundary (which is a function of the grain boundary microstructure) could prove to be useful in extending the model in its current state to incorporate damage considerations and subsequent modeling of tertiary creep behavior.