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MCFET computer program, named for combined Monte Carlo Finite Element Technique, implements specialized microstructural-lattice simulation technique. Developed to simulate microstructural evolution in material systems in which modulated phases occur and directionality of modulation influenced by internal and external stresses. Written in FORTRAN.
In 2012, a pathfinder forward pressure vessel bulkhead (FPVBH) for the Orion Multi-Purpose Crew Vehicle (MPCV) was fabricated using aluminum (Al) alloy 2219. The demonstration article was offered as a replacement for the baseline configuration, which at that time was a multi-piece welded construction using aluminum-lithium (Al-Li) alloy 2195. In 2014, the NASA Engineering and Safety Center (NESC) funded a Phase I effort to explore spin-forming as a manufacturing method to produce a complex-shaped, single-piece FPVBH for the crew module (CM) [1]. The primary focus of the NESC effort was to expand spin-forming technology to the fabrication of an Al-Li 2195 FPVBH. As a result, limited testing and analysis was performed on the original Al 2219 article. In this study, a sufficiently large plate of Al 2219 was not available for the fabrication of a FPVBH. Therefore, the forming blank comprised two plates (from different lots) butt welded together using a single friction stir weld. The welded forming blank was then spin formed into the FPVBH configuration and heat treated to the T62 temper. Mechanical property testing of the spin-formed 2219-T62 FPVBH revealed that the tensile properties were comparable to the Metallic Material Properties Development and Standardization (MMPDS) A-basis allowable for Al 2219-T6 wrought products. However, post-spin forming application of the standard heat treatment produced large, recrystallized grains in the material. The response varied within the two plates; one plate lot exhibited nearly complete recrystallization, while the other lot contained bands of recrystallization. Although no reduction in tensile properties was observed due to these microstructural variations, a more detailed investigation was recommended to better understand the effect of spin forming on the recrystallization mechanism and the impact on other material properties. The current study examined the microstructure and mechanical properties of the fully processed Al 2219-T62 FPVBH with emphasis on comparing differences between the two plates. Testing was limited to the cone region of the FPVBH. The two specific tasks in this study were: a) Characterization of the microstructure and mechanical properties in the two plates used in the FPVBH. b) Isolation of the impact of each step in the thermo-mechanical processing (TMP) sequence of the FPVBH on recrystallization and grain growth. Mechanical property testing sampled regions containing both large, recrystallized and small, unrecrystallized grains. Tensile strengths and elongations were equivalent in both regions and exceeded MMPDS A-basis allowables. Fracture toughness was equivalent in both regions and fracture morphology was typical of aluminum alloys. The fatigue precrack region of the fracture toughness specimens appeared more faceted in the recrystallized plate. This suggests that property testing that is more sensitive to grain size, such as fatigue crack growth rate and stress corrosion cracking, may be warranted. Alcoa reviewed the production records for the two plate lots and confirmed that composition of both plates was within the allowable range for Al 2219 and that plate processing steps were all performed within production limits. However, the Fe content was notably higher in the fully recrystallized plate, resulting in a greater volume fraction of large Fe-bearing inclusions and Al-Cu constituent particles. Alcoa did not provide details regarding plate processing variables for the two plate lots. It was surmised that deformation by hot rolling was the most accurate TMP simulation of commercial spin forming within a laboratory setting. Both hot rolling and spin forming comprise combinations of shear and compressive deformation. The through-thickness strain caused by hot rolling is symmetrical about the mid-plane, as the rollers contact both sides of the material. Shear strains are maximum at the inner and outer surfaces and trend toward zero at the mid-thickness. In contrast, spin forming utilizes a single roller on the outer surface, producing a gradient in shear stress through the thickness. Shear strains in spin-formed material are highest at the outer surface due to direct contact with the roller, and trend toward zero at the inner surface. Consequently, there will undoubtedly be some discrepancies in the mechanical response of hot-rolled and spin-formed products in the through-thickness direction. Interrupted TMP simulations were performed on remnant plate from the FPVBH forming blank in order to isolate the effects of a post-weld anneal, spin-forming deformation/thermal cycling, and solution heat treatment (SHT) on development of the recrystallized microstructures. Analysis confirmed that large, recrystallized grains formed exclusively during SHT and only after a certain deformation level was exceeded. Remnant plate from both lots that received thermal processing only did not exhibit recrystallization but exhibited fully recrystallized microstructures when processed by hot rolling. This indicates that (1) the level of deformation exceeded the critical level for recrystallization in both plates and (2) that the level of deformation during hot rolling was greater than that which accumulated during spin forming of the FPVBH. Comparison of the hot-rolled plate with the spin-formed material showed that the recrystallized grain size was larger in the FPVBH, providing further evidence that deformation levels were lower in the FPVBH. The most plausible explanation for the differing degrees of recrystallization in the two plates is particle stimulated nucleation (PSN) at the Fe-bearing inclusions and large Al-Cu constituent particles. In the plate with higher solute content, the greater volume fraction of these particles reduces the deformation level required to promote recrystallization. During spin forming the critical deformation level was likely exceeded for the higher Fe content plate resulting in a fully recrystallized microstructure. The lower Fe content plate developed a microstructure that exhibited bands of recrystallization due to the combination of a non-uniform distribution of particles and deformation. The critical deformation required for recrystallization was likely only exceeded during spin forming in regions of higher particle content. While there were differences in Fe content between these lots of Al 2219 plate, the composition of both was within defined alloy limits and Alcoa reported no anomalies during plate production. The observed differences in recrystallization after spin forming may reflect the inherent variability between these two lots of plate. The spin-forming vendor, Spincraft, reported that similar variations in recrystallized microstructures have been observed during examination of multiple spin-forming trials of a different 2xxx series aluminum alloy. Hence, tighter chemical specification limits on impurity levels in Al 2219 plate may help ensure uniform, predictable microstructures. Finally, the tensile and fracture toughness properties were not affected by the microstructural variations noted.
Repeated plastic deformation, which of course depends on dislocation mechanism, at or near the crack tip leads to the fatigue crack propagation. By involving the theory of thermally activated flow and the cumulative plastic strain criterion, an effort is made here to model the stage II fatigue crack propagation rate in terms of the dislocation mechanism. The model, therefore, provides capability to ascertain: (1) the dislocation mechanism (and hence the near crack tip microstructures) assisting the crack growth, (2) the relative resistance of dislocation mechanisms to the crack growth, and (3) the fracture surface characteristics and its interpretation in terms of the dislocation mechanism. The local microstructure predicted for the room temperature crack growth in copper by this model is in good agreement with the experimental results taken from the literature. With regard to the relative stability of such dislocation mechanisms as the cross-slip and the dislocation intersection, the model suggests an enhancement of crack growth rate with an ease of cross-slip which in general promotes dislocation cell formation and is common in material which has high stacking fault energy (produces wavy slips). Cross-slip apparently enhances crack growth rate by promoting slip irreversibility and fracture surface brittleness to a greater degree.
Powder bed fusion (PBF) additive manufacturing has found numerous applications in the aerospace domain. However, components fabricated via PBF have a complex time-temperature history that significantly impacts subsequent mechanical performance. This study examines convolution-based numerical solutions of transient temperature fields that support simulations involving arbitrary beam shapes and paths during PBF. The convolutional approach is verified through comparisons with analytical solutions of the temperature field. The computational speed and accuracy of the method are assessed through comparisons with other explicit and implicit numerical techniques. In addition, the straightforward translation of the approach from a CPU to a GPU implementation and the resultant performance improvement are presented. The role of the technique in predicting microstructure evolution during PBF (for a greater process-structure-property-performance framework) is also demonstrated. This work supports the development of computational materials methods for understanding and controlling the time-temperature history during PBF.
Electron beam freeform fabrication (EBF3) is a member of an emerging class of direct manufacturing processes known as solid freeform fabrication (SFF); another member of the class is the laser deposition process. Successful application of the EBF3 process requires precise control of a number of process parameters such as the EB power, speed, and metal feed rate in order to ensure thermal management; good fusion between the substrate and the first layer and between successive layers; minimize part distortion and residual stresses; and control the microstructure of the finished product. This is the only effort thus far that has addressed computer simulation of the EBF3 process. The models developed in this effort can assist in reducing the number of trials in the laboratory or on the shop floor while making high-quality parts. With some modifications, their use can be further extended to the simulation of laser, TIG (tungsten inert gas), and other deposition processes. A solid mechanics-based finite element code, ABAQUS, was chosen as the primary engine in developing these models whereas a computational fluid dynamics (CFD) code, Fluent, was used in a support role. Several innovative concepts were developed, some of which are highlighted below. These concepts were implemented in a number of new computer models either in the form of stand-alone programs or as user subroutines for ABAQUS and Fluent codes. A database of thermo-physical, mechanical, fluid, and metallurgical properties of stainless steel 304 was developed. Computing models for Gaussian and raster modes of the electron beam heat input were developed. Also, new schemes were devised to account for the heat sink effect during the deposition process. These innovations, and others, lead to improved models for thermal management and prediction of transient/residual stresses and distortions. Two approaches for the prediction of microstructure were pursued. The first was an empirical approach involving the computation of thermal gradient, solidification rate, and velocity (G,R,V) coupled with the use of a solidification map that should be known a priori. The second approach relies completely on computer simulation. For this purpose a criterion for the prediction of morphology was proposed, which was combined with three alternative models for the prediction of microstructure; one based on solidification kinetics, the second on phase diagram, and the third on differential scanning calorimetry data. The last was found to be the simplest and the most versatile; it can be used with multicomponent alloys and rapid solidification without any additional difficulty. For the purpose of (limited) experimental validation, finite element models developed in this effort were applied to three different shapes made of stainless steel 304 material, designed expressly for this effort with an increasing level of complexity. These finite element models require large computation time, especially when applied to deposits with multiple adjacent beads and layers. This problem can be overcome, to some extent, by the use of fast, multi-core computers. Also, due to their numerical nature coupled with the fact that solid mechanics- based models are being used to represent the material behavior in liquid and vapor phases as well, the models have some inherent approximations that become more pronounced when dealing with multi-bead and multi-layer deposits.
Casting and welding of superalloys, stainless steel and titanium alloys are processes which can be improved through modeling of heat flow, fluid flow, residual stress development, and microstructural evolution. These simulations require inputs of thermophysical data, some of which involves the partially or totally liquid state. In particular, these processes involve melting, flow in the liquid, and solidification. Modeling of such processes can lead to an improved understanding of defects such as shrinkage, inclusions, cracks, incomplete filling (or penetration), macrosegregation, improper grain structure, and deviations from dimensional specifications. Effective modeling can shorten process development time and improve quality. An approach to these problems is to develop efficient models; validate through correlations with thermal, distortion, and microstructural data; run parametric studies; extract knowledge based rules; and apply to adaptive closed loop control systems. With the appropriate pre- and post-processing, such analyses can be made 'user friendly'. This would include graphical user interfaces as well as realistic images and color maps. In such form, these models can be used for sensitivity analyses, which are useful in defining appropriate sensors and in the development of control strategies. Such modeling can be done at several levels, e.g., the MARO level, modeling large scale phenomena such as heat and fluid flow or material deformation; the MICRO level, modeling the development of dendrites, grains or precipitates; or at the NANO level, modeling point defects, dislocations, stacking faults, etc. There are many computational issues associated with these simulations, e.g. computational efficiency and accuracy. In addition, there are many materials issues, not the least of which is the availability of accurate high temperature thermophysical data for complex alloys. This would include latent heat of fusion, temperature dependent heat capacity and thermal conductivity (for liquid and solid), viscosity, surface tension, thermal expansion, mechanical properties, etc. Preliminary data is frequently gathered from the literature; however, this is often not available for modern alloys. If additional data are required, measurements can be used; however, these are costly, time consuming and can be erroneous due to a lack of testing standards or impure materials. Microstructural predictors can be extracted from thermal information, e.g. cooling rate and thermal gradient; the prediction of microstructure is dependent on solidus and liquidus temperature, mushy zone permeability, the solidification curve, volume changes, phase transformations, alloying effects (such as surface tension or viscosity), mold/metal reactions, metal/environment reactions, etc. Defect maps may be needed to predict the onset of shrinkage, hot cracking or 'freckling'. Constants may be needed for stress relaxation, dendrite coarsening, vaporization, etc. Visualization was used as a tool to better comprehend complex data sets associated with the analysis of directional solidification (including crystal growth) and welding. Examples include not only isotherms, but also cooling rate, growth rate and thermal gradient. The latter two are not single valued scalars, but rather time and space dependent vector fields. Efficient models were developed for both casting and welding to predict heat flow and the relationship to dendrite and grain growth. These codes include many of the non-linear effects, e.g. radiation, which dominate these processes. The home-built FDM code(s) were designed to be useful not only to the scientist, but also to the process engineer. Special output can be requested to compare directly to experimental data. Visualization procedures were developed to visualize critical results, e.g. fusion zone width at the surface opposite that where the arc is applied ('penetration'). Both elaborate and simplified distortion analyses were carried out. It is clear that extensive mechanical property data are critical in order to accurately predict residual stress patterns. A scheme is currently being developed to integrate these modeling tools into a set of control algorithms; however, the success of this approach is critically dependent on the availability of accurate high temperature thermophysical data.
The objective of this initiative, funded by NASA's Aviation Safety Program, is to model, validate, and predict, with high fidelity, the microstructural evolution of third-generation high-refractory Ni-based disc superalloys during heat treating and service conditions. This initiative is a natural extension of the DARPA-AIM (Accelerated Insertion of Materials) initiative with GE/Pratt-Whitney and with other process simulation tools. Strong collaboration with the NASA Glenn Research Center (GRC) is a key component of this initiative and the focus of this program is on industrially relevant disk alloys and heat treatment processes identified by GRC. Employing QuesTek s Computational Materials Dynamics technology and PrecipiCalc precipitation simulator, physics-based models are being used to achieve high predictive accuracy and precision. Combining these models with experimental data and probabilistic analysis, "virtual alloy design" can be performed. The predicted microstructures can be optimized to promote desirable features and concurrently eliminate nondesirable phases that can limit the reliability and durability of the alloys. The well-calibrated and well-integrated software tools that are being applied under the proposed program will help gas turbine disk alloy manufacturers, processing facilities, and NASA, to efficiently and effectively improve the performance of current and future disk materials.
The objective of this program was to model, validate, and predict the precipitation microstructure evolution, using PrecipiCalc (QuesTek Innovations LLC) software, for 3rd generation Ni-based gas turbine disc superalloys during processing and service, with a set of logical and consistent experiments and characterizations. Furthermore, within this program, the originally research-oriented microstructure simulation tool was to be further improved and implemented to be a useful and user-friendly engineering tool. In this report, the key accomplishments achieved during the third year (2009) of the program are summarized. The activities of this year included: Further development of multistep precipitation simulation framework for gamma prime microstructure evolution during heat treatment; Calibration and validation of gamma prime microstructure modeling with supersolvus heat treated LSHR; Modeling of the microstructure evolution of the minor phases, particularly carbides, during isothermal aging, representing the long term microstructure stability during thermal exposure; and the implementation of software tools. During the research and development efforts to extend the precipitation microstructure modeling and prediction capability in this 3-year program, we identified a hurdle, related to slow gamma prime coarsening rate, with no satisfactory scientific explanation currently available. It is desirable to raise this issue to the Ni-based superalloys research community, with hope that in future there will be a mechanistic understanding and physics-based treatment to overcome the hurdle. In the mean time, an empirical correction factor was developed in this modeling effort to capture the experimental observations.
Interdiffusion in Ni-rich Ni-Cr-Al alloys is investigated experimentally after annealing at 1100 and 1200 C using gamma/gamma, gamma/gamma+beta, gamma/gamma+gamma prime, and gamma/gamma+alpha diffusion couples. The amount and location of Kirkendall porosity suggests that Al diffuses more rapidly than Cr which diffuses more rapidly than Ni in the gamma phase of Ni-Cr-Al alloys. The location and extent of maxima and minima in the concentration profiles of the diffusion couples indicate that both cross-term diffusion coefficients are positive. Measurements are also presented of the ternary interdiffusion coefficients of the gamma phase in the Ni-Cr-Al system. It is shown that the interdiffusion coefficients can be accurately predicted by using a ternary finite-difference interdiffusion model.
The microstructural evolution of a commercial single crystal superalloy, NASAIR 100, is analyzed using the existing high-temperature lattice mismatch data and high-temperature moduli obtained from tests on single crystals of gamma and gamma prime. A multiparticle analysis of the microstructural evolution is performed using a novel microstructural lattice simulation technique, MCFET. Under a uniaxial stress, a regular array of gamma prime particles in the simulated microstructure is predicted to coalesce and form a plate morphology, with the broad faces of the plates and stress axis perpendicular in tension but parallel in compression. These results are consistent with changes in gamma prime shape observed in NASAIR 100 following creep testing at 1000 C.
An experimental study by holographic interferometry is reported of the following material properties of conventional and negative Poisson's ratio copper foams: Young's moduli, Poisson's ratios, yield strengths and characteristic lengths associated with inhomogeneous deformation. The Young's modulus and yield strength of the conventional copper foam were comparable to those predicted by microstructural modeling on the basis of cellular rib bending. The reentrant copper foam exhibited a negative Poisson's ratio, as indicated by the elliptical contour fringes on the specimen surface in the bending tests. Inhomogeneous, non-affine deformation was observed holographically in both foam materials.
Flexible manufacturing methods are needed to reduce the cost of using advanced composites in structural applications. One method that allows for this is the stretch forming of long discontinuous fiber materials with thermoplastic matrices. In order to exploit this flexibility in an economical way, a thorough understanding of the relationship between manufacturing and component performance must be developed. This paper reviews some of the recent work geared toward establishing this understanding. Micromechanics models have been developed to predict the formability of the material during processing. The latest improvement of these models includes the viscoelastic nature of the matrix and comparison with experimental data. A finite element scheme is described which can be used to model the forming process. This model uses equivalent anisotropic viscosities from the micromechanics models and predicts the microstructure in the formed part. In addition, structural models have been built to account for the material property gradients that can result from the manufacturing procedures. Recent developments in this area include the analysis of stress concentrations and a failure model each accounting for the heterogeneous material fields.
Direct Energy Deposition (DED) is an emerging technology extensively employed in metal Additive Manufacturing (AM). Despite its widespread industrial application, mathematical modeling in this domain remains challenging. This complexity arises from the intricate nature of the modeling approach and the nonlinear behavior of material parameters across a broad temperature range. Consequently, experimentalists often resort to a trial-and-error method to achieve structures with desired properties. However, this approach can be time-consuming and may not always yield parts with the requisite characteristics, highlighting the necessity for mathematical modeling to enhance manufacturing success. This study focuses on thin-walled manufacturing with a single bead thickness, adjusting laser parameters to produce such structures. The laser cladding speed, typical for DED manufacturing, is considered to be on the order of centimeters per second. The various wall thicknesses and heights are explored to discern the general characteristics of walls manufactured via this DED approach. Our modeling approach diverges from the conventional DED modeling based on activation-deactivation of predefined mesh domains, commonly implemented in many finite element codes. Instead, a method is proposed that effectively models layer cladding and melt pool dynamics, enabling predictions of the microstructure in the resulting structures. The formation mechanisms of cellular, dendritic columnar, and stray (equiaxed) grains, which arise from the interplay between nucleation and growth from the surface is analyzed. The study also examines the feasibility of microstructure formation to demonstrate the various thermal characteristics inherent in wall manufacturing. Our results illustrate how different process parameters influence the temperature gradient and cooling rate of the molten pool, subsequently affecting the primary dendrite arm spacing (PDAS). This modeling technique allows for the investigation of diverse thermal conditions, facilitating the prediction of microstructure and residual stresses in the manufactured parts.
The objective of this program is to model, validate, and predict the precipitation microstructure evolution, using PrecipiCalc (QuesTek Innovations LLC) software, for 3rd generation Ni-based gas turbine disc superalloys during processing and service, with a set of logical and consistent experiments and characterizations. Furthermore, within this program, the originally research-oriented microstructure simulation tool will be further improved and implemented to be a useful and user-friendly engineering tool. In this report, the key accomplishment achieved during the second year (2008) of the program is summarized. The activities of this year include final selection of multicomponent thermodynamics and mobility databases, precipitate surface energy determination from nucleation experiment, multiscale comparison of predicted versus measured intragrain precipitation microstructure in quench samples showing good agreement, isothermal coarsening experiment and interaction of grain boundary and intergrain precipitates, primary microstructure of subsolvus treatment, and finally the software implementation plan for the third year of the project. In the following year, the calibrated models and simulation tools will be validated against an independently developed experimental data set, with actual disc heat treatment process conditions. Furthermore, software integration and implementation will be developed to provide material engineers valuable information in order to optimize the processing of the 3rd generation gas turbine disc alloys.
Solidification microstructures that form under steady-state growth conditions (cells, dendrites, regular eutectics, etc.) are reasonably well understood in comparison to other, more complex microstructures, which form under intrinsically non-steady-state growth conditions due to the competition between the nucleation and growth of several phases. Some important practical examples in this latter class include microstructures forming in peritectic systems in highly undercooled droplets, and in strip cast stainless steels. Prediction of phase and microstructure selection in these systems has been traditionally based on (1) heterogeneous nucleation on a static interface, and (2) comparing the relative growth rate of different phase/microstructures under steady-state growth conditions. The formation of new phases, however, occurs via nucleation on, or ahead of, a moving boundary. In addition, the actual selection process is controlled by a complex interaction between the nucleation process and the growth competition between the nuclei and the pre-existing phase under non-steady-state conditions. As a result, it is often difficult to predict which microstructure will form and which phases will be selected under prescribed processing conditions. This research addresses this critical role of nucleation at moving boundaries in the selection of phases and solidification microstructures through quantitative experiments and numerical modeling in peritectic systems. In order to create a well characterized system in which to study this problem, we focus on the directional solidification of hypo- and hyper-peritectic alloys in the two-phase region, imposing a large enough ratio of temperature gradient/growth rate (G/V(sub p)) to suppress the morphological instability of both the parent (alpha) and peritectic (Beta) phases, i.e. each phase alone would grow as a planar front. Our combined experimental and theoretical results show that, already in this simplified case, the growth competition of these two phases leads to a rich variety of microstructures that depend sensitively upon the relative importance of nucleation, diffusion, and convection.
The objective of this paper is to develop further a framework for computationally modeling microstructurally small fatigue crack growth in AA 7075-T651 [1]. The focus is on the nucleation event, when a crack extends from within a second-phase particle into a surrounding grain, since this has been observed to be an initiating mechanism for fatigue crack growth in this alloy. It is hypothesized that nucleation can be predicted by computing a non-local nucleation metric near the crack front. The hypothesis is tested by employing a combination of experimentation and nite element modeling in which various slip-based and energy-based nucleation metrics are tested for validity, where each metric is derived from a continuum crystal plasticity formulation. To investigate each metric, a non-local procedure is developed for the calculation of nucleation metrics in the neighborhood of a crack front. Initially, an idealized baseline model consisting of a single grain containing a semi-ellipsoidal surface particle is studied to investigate the dependence of each nucleation metric on lattice orientation, number of load cycles, and non-local regularization method. This is followed by a comparison of experimental observations and computational results for microstructural models constructed by replicating the observed microstructural geometry near second-phase particles in fatigue specimens. It is found that orientation strongly influences the direction of slip localization and, as a result, in uences the nucleation mechanism. Also, the baseline models, replication models, and past experimental observation consistently suggest that a set of particular grain orientations is most likely to nucleate fatigue cracks. It is found that a continuum crystal plasticity model and a non-local nucleation metric can be used to predict the nucleation event in AA 7075-T651. However, nucleation metric threshold values that correspond to various nucleation governing mechanisms must be calibrated.
A nondimensional model of microstructurally short crack growth in creep is developed based on a detailed observation of the creep fracture process of 304 stainless steel. In order to deal with the scatter of small crack growth rate data caused by microstructural inhomogeneity, a random variable technique is used in the model. A cumulative probability of the crack length at an arbitary time, G(bar a, bar t), and that of the time when a crack reaches an arbitary length, F(bar t, bar a), are obtained numerically by means of a Monte Carlo method. G(bar a, bar t), and F(bar t, bar a) are the probabilities for a single crack. However, multiple cracks generally initiate on the surface of a smooth specimen from the early stage of creep life to the final stage. TAking into account the multiple crack initiations, the actual crack length distribution observed on the surface of a specimen is predicted by the combination of probabilities for a single crack. The prediction shows a fairly good agreement with the experimental result for creep of 304 stainless steel at 923 K. The probability of creep life is obtained from an assumption that creep fracture takes place when the longest crack reaches a critical length. The observed and predicted scatter of the life is fairly small for the specimens tested.
A nondimensional model of microstructurally short crack growth in creep is developed based on a detailed observation of the creep fracture process of 304 stainless steel. In order to deal with the scatter of small crack growth rate data caused by microstructural inhomogeneity, a random variable technique is used in the model. A cumulative probability of the crack length at an arbitrary time, G(bar a, bar t), and that of the time when a crack reaches an arbitrary length, F(bar t, bar a), are obtained numerically by means of a Monte Carlo method. G(bar a, bar t), and F(bar t, bar a) are the probabilities for a single crack. However, multiple cracks generally initiate on the surface of a smooth specimen from the early stage of creep life to the final stage. Taking into account the multiple crack initiations, the actual crack length distribution observed on the surface of a specimen is predicted by the combination of probabilities for a single crack. The prediction shows a fairly good agreement with the experimental result for creep of 304 stainless steel at 923 K. The probability of creep life is obtained from an assumption that creep fracture takes place when the longest crack reaches a critical length. The observed and predicted scatter of the life is fairly small for the specimens tested.