9.6analysis procedures for dynamic and time dependent properties

MIL-HDBK-5J, Metallic Materials and Elements for Aerospace Vehicle Structures

Last Updated 29 Oct 2026

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9.6.1

LOAD AND STRAIN CONTROL FATIGUE DATA — Fatigue has been defined as “the process of progressive localized permanent structural change occurring in a material subjected to conditions that produce fluctuating stresses and strains at some point or points, and which may culminate in cracks or complete fracture after a sufficient number of fluctuations.”

For many years, tests have been performed on specimens having simple geometries in attempts to characterize the fatigue properties of particular materials. Fatigue tests have been conducted for many reasons. Basic fatigue-life information may be desired for design purposes, or to evaluate the differences between materials. The effects of heat treatments, mechanical working, or material orientation may also be studied through comparative fatigue testing.

Many types of machines and specimen designs have been used to develop fatigue data. Machine types include mechanical, electromechanical, hydraulic, and ultrasonic. Specimens have been designed for testing in cyclic tension and/or compression, bending, and torsion. Cyclic loading conditions have been produced by rotating bending, axial loading and cantilever bending. In- and out-of-phase biaxial and multiaxial fatigue conditions have also been examined using specially designed specimens. Tests have been conducted in a variety of simulated environments including temperatures ranging from cryogenic to near melting point levels. The fatigue data included in MIL-HDBK-5 are limited to constant-amplitude axial fatigue data on simple laboratory specimens tested according to ASTM E 606. Data obtained under both strain control and load (stress) control are included. Figure 9.6.1(a) shows examples of trends for stress-life and strain-life fatigue data. Generally, stress-life data for unnotched specimens are limited to stress levels that produce intermediate-to-long fatigue lives because of unstable cyclic creep and tensile failure that can occur at high stress ratios in load-control testing. This phenomenon is shown in Figure 9.6.1(b). Strain-life curves are often focused on strain ranges that produce short-to-intermediate fatigue lives because of strain rate and frequency limitations which require long testing times to generate long-life fatigue data under strain control. However, there is no inherent limit to the life range that can be evaluated in strain-control testing.

For fatigue to occur, a material must undergo cyclic plasticity, at least on a localized level. The relationship between total strain, plastic strain, and elastic strain is shown in Figure 9.6.1(c). Low-cycle fatigue tests involve relatively high levels of cyclic plasticity. Intermediate-life fatigue tests usually involve plastic strains of the same order as the elastic strains. Long-life fatigue tests normally involve very low levels of cyclic plasticity. These trends are shown in Figure 9.6.1(d). In the MIL-HDBK-5 fatigue analysis guidelines, engineering strain is denoted as e and true or local strain is denoted as $\varepsilon$ . These symbols are used interchangeably within MIL-HDBK-5 for small strain values.

The limited plasticity involved in intermediate and long-life fatigue tests often results in a similar stress-strain response for both fully reversed strain-control and fully reversed load-control tests. A fatigue test, under strain control that produces a stable maximum stress of X, should produce (on the average) a fatigue life that is comparable to that obtained for a sample tested under load control at a maximum stress of X. Strictly speaking, the results are likely to be most comparable in terms of crack initiation life and not total life. If the comparison is made in terms of total life, the load-control results will tend to be more conservative than those generated by strain-control testing. When a specimen cracks in a test under strain control, it will usually display a decrease in maximum tensile load. Under load control, the maximum tensile load will remain constant but stress will increase as the crack grows, resulting in a shorter period of crack growth before the specimen fails.

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9.6.1.1

Data Collection and Interpretation — If a set of strain- or load-control data for a material of interest meet the minimum requirements, the data should be processed for analysis. Load-control data reports should clearly specify the net section stresses, stress ratios, and associated cycles to failure. Strain-control data reports should clearly specify the strain levels used, the stable stress response values, and the associated cycles to initiation and/or failure, along with a clear and concise definition of the failure criterion. Acceptable definitions of failure in a strain-control fatigue test report include:

(1) Total specimen separation

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9.6.1.2

Analysis of Data — Once a collection of data is reviewed (see Section 9.6.1.1) and compiled for the material of interest, analysis of that data may begin. An outline of the analysis procedure that is normally followed is given in Figure 9.6.1.2. Each of the elements in the flow chart are discussed in the following sections.

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9.6.1.3

Fatigue Life Models — To clarify the fatigue data trends for a specific stress or strain ratio, a linear regression model can be applied as follows:

\log \left(\mathrm{N}_{\mathrm{i}} \text { or } \mathrm{N}_{\mathrm{f}}\right)=\mathrm{A}_{1}+\mathrm{A}_{2} \log \left(\mathrm{S}_{\max } \text { or } \Delta \epsilon\right) .

[9.6.1.3(a)]

Note that fatigue life is specified as the dependent variable. The alternative approach, using stress or strain as the dependent variable, is sometimes used, but this procedure will not be employed in developing mean fatigue curves in MIL-HDBK-5. The use of fatigue life or, more specifically, logarithm (base 10) of fatigue life as the dependent variable will be used since stress or strain is the controlled parameter in a fatigue experiment, and the resultant fatigue life is a random variable.

If Equation 9.6.1.3(a) does not adequately describe long-life data trends, a nonlinear model (or a more complicated linear model) may be warranted. For example, long-life, load-control data might be modeled by the nonlinear expression

\log \mathrm{N}_{\mathrm{j}}=\mathrm{A}_{1}+\mathrm{A}_{2}\left(\mathrm{S}_{\max }-\mathrm{A}_{3}\right)

[9.6.1.3(a)]

or by the more complicated equation [Reference 9.6.1.3]

\log N_{\mathrm{f}}=A_{1}+A_{2} \log S_{\max }+A_{3} \sqrt{\log S_{\max }}+A_{4}

[9.6.1.3(c)]

These more complex forms should only be employed in instances where they are warranted based on a distinct fatigue limit at long lives and when the simpler linear model was inadequate.

Standard least squares regression analysis and the procedure for detecting outliers in Section 9.6.1.6 require that the variance be relatively constant at all fatigue life values. Traditionally, the logarithm of fatigue life is approximated by a normal distribution. However, the variability or scatter of fatigue life is generally not constant, but increases with increasing fatigue life. To ensure the reliable use of the outlier detection procedure, a weighting scheme designed to produce a more uniform distribution of residuals is suggested in Section 9.6.1.5.

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9.6.1.4

Evaluation of Mean Stress and Strain Effects—Commonly, load-controlled fatigue data generated over a range of stress ratios can be represented by the following equivalent stress-fatigue life formulation:

\log \mathrm{N}_{\mathrm{f}}=\mathrm{A}_{1}+\mathrm{A}_{2} \log \left(\mathrm{S}_{\mathrm{eq}}-\mathrm{A}_{4}\right)

[9.6.1.4(a)]

where

$\mathrm{S}_{\mathrm{eq}}=\left(\Delta \mathrm{S}\right)^{\mathrm{A}_{3}}\left(\mathrm{S}_{\max }\right)^{1-\mathrm{A}_{3}}$

$\mathrm{S}_{\mathrm{eq}}=\mathrm{S}_{\max }\left(1-\mathrm{R}\right)^{\mathrm{A}_{3}}$

The equivalent stress model (and the related equivalent strain model) are derived from Reference 9.6.1.4(a).

Equation 9.6.1.4(a) is nonlinear in its general form and must, therefore, normally be optimized through use of a nonlinear regression package. However, the above equation can be solved through a linear analysis, if A3 and A4 are optimized through an iterative solution. The parameter A3 normally lies in the range of 0.30 to 0.70, while A4 represents, in essence, the fatigue limit stress. In cases where the optimum value of A4 is negative or insignificant, it should be omitted. Unnotched data, especially aluminum alloy data, can frequently be represented without using the nonlinear A4 term. Parameter optimization is discussed more thoroughly in Section 9.6.1.5.

If $\mathrm{A}_{4}$ is zero or set equal to zero, Equation 9.6.1.4(a) becomes linear in $\log \mathrm{S}_{\max }$ and $\log (1-\mathrm{R})$ , and it can be written as follows:

$\log \mathrm{N}{\mathrm{f}}=\mathrm{A}{1}+\mathrm{A}{2} \log \mathrm{S}{\max }+\mathrm{B} \log (1-\mathrm{R}) \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \

where ( \mathrm{B}=\mathrm{A}{2} \mathrm{~A}{3} $. Thus, if$ \mathrm{A}_{4}$ is zero, then

$\mathrm{A}_{3}=\mathrm{B} / \mathrm{A}_{2}$

Strain-controlled fatigue data generated over a range of strain ratios often can be consolidated by the following equivalent strain formulation:

\log \mathrm{N}_{\mathrm{f}}=\mathrm{A}_{1}+\mathrm{A}_{2} \log \left(\epsilon_{\mathrm{eq}}-\mathrm{A}_{4}\right)

[9.6.1.4(c)]

where

$\boldsymbol{\epsilon}_{\mathrm{eq}}=\left(\Delta \boldsymbol{\epsilon}\right)^{A_{3}}\left(\mathrm{S}_{\max } / \mathrm{E}\right)^{1-A_{3}}$

Note that Equation 9.6.1.4(c) is very similar in form to Equation 9.6.1.4(a). It is important to note, however, that the maximum stress value used in Equation 9.6.1.4(c) is not a controlled quantity. It is a measured quantity and its magnitude depends primarily on the amount of cyclic softening or hardening that occurs in combination with mean stress relaxation. Although $S_{\max }$ can be predicted with reasonable accuracy if the cyclic response of the material is well established, using the stable measured values of $S_{\max }$ , when analyzing strain-control data for presentation in MIL-HDBK-5, is preferred.

The equivalent stress and strain approaches are very useful for computing mean fatigue life estimates for conditions intermediate to those for which the test data have been generated. Caution should be used, however, in making life predictions for stress/strain conditions beyond the range of those represented in the

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9.6.1.5

Estimation of Fatigue-Life Model Parameters — The fatigue-life model parameters are estimated to obtain the best-fit S/N or ε/N curve for the data. The procedure used to determine the parameters includes a statistical method for adjusting the fatigue model for the nonconstant variance commonly observed in long-life fatigue data. The motivation for this adjustment is the fact that constant variance is an inherent assumption in least squares regression analysis. To estimate the parameters in Equation 9.6.1.4(a) or Equation 9.6.1.4(c) and adjust the model to incorporate nonuniform variance, the following six-step procedure is performed.

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9.6.1.6

Treatment of Outliers — An outlying observation (or outlier) is one that appears to deviate markedly from other observations in the sample in which it occurs. Outliers may essentially be classified into two groups:

(1) An extreme value of the random variable inherent in the data (in this case fatigue life). If this is true, the value should be retained in future analyses.
(2) An unusual result caused by a gross deviation in material or prescribed experimental procedure or an error in calculating or recording any experimental data.

An outlier of the second type is sometimes correctable by a review of the test sample and/or test records, which may provide sufficient evidence for rejection of the observation. An outlying value from a failure that occurred in the fillet of an unnotched fatigue test sample is an example of a potentially rejectable result based on physical evidence alone. The more difficult case is one where an observation is an obvious outlier and no physical reasons can be identified to justify its exclusion.

Assuming uniform variance in the standardized residuals over the complete range in equivalent stress or strain, the problem of identifying certain observations as potential outliers should be addressed as follows. Calculate the studentized residuals,

\mathrm{T}_{\mathrm{i}}=\frac{\mathrm{SR}_{\mathrm{i}}}{\left(1-\mathrm{h}_{\mathrm{i}}\right)^{1 / 2}}\left[\frac{\mathrm{RMSE}}{\mathrm{RMSE}(\mathrm{i})}\right]

[9.6.1.6(a)]

for $i=1, \ldots, n$ where $\mathrm{SR}_{i}$ is the standardized residual from Equation 9.6.1.5(k), RMSE is the root mean square error based on the entire sample as calculated in either Equation 9.6.1.5(e) or Equation 9.6.1.5(i), and RMSE(i) is the root mean square error based on the sample which excludes the ith observation as calculated by either Equation 9.6.1.5(e) or Equation 9.6.1.5(i).

The value $\mathrm{h}_{\mathrm{i}}$ is calculated using the formula

=\frac{\mathrm{X}_{\mathrm{li}}^{2}\left(\sum \mathrm{X}_{2 \mathrm{j}}^{2}\right)-2 \mathrm{X}_{\mathrm{li}} \mathrm{X}_{2 \mathrm{i}}\left(\sum \mathrm{X}_{1 \mathrm{j}} \mathrm{X}_{2 \mathrm{j}}\right)+\mathrm{X}_{2 \mathrm{j}}^{2}\left(\sum \mathrm{X}_{1 \mathrm{j}}^{2}\right)}{\left(\sum \mathrm{X}_{2}^{2}\right)\left(\sum \mathrm{X}_{1}^{2}\right)-\left(\sum \mathrm{X}_{1} \mathrm{X}_{2}^{2}\right)}

[9.6.1.6(b)]

where $X_{1 \mathrm{i}}$ is the value of $1 / \mathrm{SD}$ for the ith specimen, $X_{2 \mathrm{i}}$ is the value of $\log \left(\mathrm{S}_{\mathrm{eq}}-\mathrm{A}_{4}\right) / \mathrm{SD}$ for the ith specimen and all summations are over $\mathrm{j}=1, \ldots, \mathrm{n}$ . Note that

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9.6.1.7

Assessment of the Fatigue Life Model — The fit of the fatigue model S/N curve to the data may be assessed in two ways—the adequacy of the equivalent stress/strain model and the adequacy of the fatigue life model. The equivalent stress model lack of fit test and the overall lack of fit test described below provide a reasonable assessment of the fatigue life model.

When three or more stress (strain) ratios are used, the fit of the equivalent stress (strain) model may be tested by determining the relationship between the standardized residuals from Equation 9.6.1.5(k) and stress (strain) ratio. A difference in the means of the standardized residuals at each stress (strain) ratio indicates that the equivalent stress (strain) model is inadequate. To determine whether or not there is a statistically significant difference in the means of the standardized residuals at each stress (strain) ratio, an

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9.6.1.8

Data Set Combination — In many cases, data from different sources, orientations, etc., may need to be combined for analysis. When data set combinations of this sort are performed, the validity of the combination should be tested with the method described below. The test is similar to that used to determine the adequacy of the equivalent stress (strain) model in the previous section.

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9.6.1.9

Treatment of Runouts — It is difficult to incorporate information from runouts (or interrupted tests) when using the least squares criterion to fit fatigue life models to data since the failure times for these observations are not known. The runouts must be either ignored or treated as failures and neither of these alternatives adequately incorporates the information contained in the runout observations. Both of these approaches tend to produce smaller predicted lives at a given equivalent stress (strain) value than is appropriate. The treatment of runouts presented below is more appropriate but requires that two of the fatigue

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9.6.1.10

Recognition of Time Dependent Effects — All prior discussion has been based on the assumption that time dependent effects in the fatigue data sample of interest are negligible. When dealing with elevated temperature fatigue properties of materials (or room temperature fatigue properties in a corrosive environment, for example), this assumption may not be realistic. Analysis methods that are approved for use in MIL-HDBK-5 do not account for time-dependent effects. Therefore, every effort must be made to identify data that embody significant time-dependent effects.

There are no absolute methods presently available for sensing time-dependent effects in fatigue data; however, there are some useful approximation techniques. One of the more useful approaches applied to “suspect” data is to include time-dependent terms in the regression model. If the terms are significant, there is reason to believe that the population contains time dependent data. Subdividing the data into subsets that do not show time dependent effect may be possible. If this is not possible, the data set should either be rejected or included with a disclaimer restricting usage of the data to predict performance at other frequencies or temperatures.

One other possible indicator of time dependent effects is an abnormal equivalent stress (strain) model. If data for different stress or strain ratios do not fit the customary models (as described in Section 9.6.1.4), or abnormal optimum parameters are defined the problem may be caused by time dependent effects. In the case of the primary equivalent stress (strain) formulation equation the exponent normally is between zero and one. If the $\mathrm{A}_{3}$ exponent approaches or exceeds one, the influence of maximum stress on fatigue life is negligible. This is a very unusual result that usually indicates problems with the data sample. The problem may result from mixed sources, where the data from each source were generated at different stress (strain) ratios. Rejection of such data sets is discussed in Section 9.6.1.8. In the case of the primary equivalent stress model [Equation 9.6.1.4(a)], if the exponent $\left(\mathrm{A}_{3}\right)$ approaches or is less than zero, it indicates the influence of maximum stress on fatigue life is "too strong". This result implies that creep is affecting the data.

If data are available for a material at a range of different temperatures it may be possible to analyze these sets separately and make comparisons between best-fit mean trend lines for increasing temperatures. If the different mean trend lines are not consistent with the higher-temperature curves converging or diverging from the lower-temperature curves, there is probably a significant time-dependent effect in the data. The suspect data should either be excluded or included with a disclaimer as previously cited. If data are excluded for time-dependent effects, the preliminary analyses of those data should be included in the data proposal and reasons for their exclusion should be given.

9.6.2

FATIGUE CRACK GROWTH DATA — Fatigue-crack-propagation data, recorded in the form of crack-length measurements and cycle counts (a, N) can be presented as crack-growth curve drawn through the data points as shown in Figure 9.6.2(a).

Although data presented in this form indicate general trends, they are not generally useful for design purposes since a variety of stress levels, stress ratios, initial crack conditions, and environmental conditions are encountered.

It has been found convenient to model fatigue-crack-propagation damage behavior as rate process and formulate a dependent variable based on the slope of this growth curve, or an approximation to it, namely,

$\frac{\mathrm{da}}{\mathrm{dN}}\simeq\frac{\mathrm{\Delta a}}{\mathrm{\Delta N}}$ [ 9.6.2(a) ]

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9.6.2.1

Data Collection and Interpretation — Reporting of basic crack-growth data will be as complete as possible. In addition to reporting cyclic loading conditions, such as maximum cyclic load and/or stress levels, stress ratio, test frequency, and specimen dimensions, it is particularly important to identify environmental conditions associated with the tests. The number of specimens and number of respective heats should also be identified. Table 9.9.2 serves as an example of the type of information which should be available (or at least is desirable) for each collection of FCP data.

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9.6.3

FRACTURE TOUGHNESS DATA — Fracture toughness of a material is its ability to resist flaw propagation and fracture. This characteristic is a generic quality, somewhat elusive to assess quantitatively. Of several measures of fracture toughness which have evolved for appraising the sensitivity of metals to the presence of small flaws, those based on crack stress or strain analysis appear to be more meaningful for use in design applications. Significant quantification of fracture and flaw propagation behavior of high-strength metals has been achieved through the concept of stress intensity factors. Typical room-temperature values and effect-of-temperature curves for critical stress intensity factors are presented in MIL-HDBK-5 for “information only” where data are available. Basic concepts, testing considerations, and interpretations of fracture toughness are briefly described in the following subsections.

A primary factor in fracture behavior of a material is stress state, i.e., plane-stress or plane-strain. In accord with previous definitions, these stress states may be interpreted mechanically as a size or thickness effect within the material. The ideal plane-stress condition occurs in the two-dimensional $\left(\sigma_{z}=0\right)$ case, in which all stresses are restricted to one plane. Typically material loaded in plane-stress can accommodate extensive plastic deformation adjacent to the flaw prior to fracture, and at fracture exhibit a relatively high K value, as computed by a relationship such as Equation 9.6.2. At the opposite extreme is the ideal plane-strain case, in which the third dimension is essentially infinite so that bulk restraint of the material permits no out-of-plane strains. As a result, plastic deformation is restricted and the material fractures in a nearly elastic manner at a relatively low K value. In real materials, these ideal extremes can be closely approximated by "quasi" conditions of "thin" and "thick" bodies. Variation in stress intensity at fracture over these extremes, and the transition stage between, may be represented as shown previously in Figure 9.2.3.5.3(a).

9.6.3.1

Plane-Strain Fracture Toughness Data — For materials which are inherently brittle, or for structures and flaw configurations which are in triaxial tension due to their thickness or bulk restraint, quasi-plane-strain-stress conditions can be obtained in a finite-sized structural element. Triaxial stress state implicit to plane strain effectively embrittles the material by providing maximum restraint against plastic deformation. In this condition, component behavior is essentially elastic until fracture stress is reached and is readily amenable to analysis in terms of elastic fracture mechanics. This mode of fracture is frequently characteristic of the very high strength metals.

9.6.3.1.1

Data Collection and Interpretation — While a wide variety of fracture specimens are available for specified testing objectives, the notch-bend specimen and compact specimen generally offer the greatest convenience and material economics for testing. Details of recommended testing practice are presented in ASTM E399.

9.6.3.2

Plane Stress and Transitional Fracture Toughness — It is convenient to consider critical stress-intensity factor values, varying with thickness or stress state, as indices of crack-damage resistance. The stress-intensity factor can be used as a consistent measure of crack damage, not only for fracture instability, but also for other levels of crack damage severity, provided the damage is consistently specified and detected. This concept implies that plane-stress and transitional-fracture toughness of metallic materials, while not necessarily a fixed value for the material, is a characteristic value for a given product form, thickness, grain direction, temperature, and strain rate.

9.6.3.2.1

Data Collection and Interpretation — Because of the complexity of crack behavior in plane-stress and transitional-stress states, test methods for evaluating material toughness have not been completely standardized; however, several useful methods do exist. Although each configuration generates nearly consistent results when data are properly evaluated, it is recommended that each general flaw configuration be interpreted and applied within its own design context.

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9.6.3.2.2

Analysis of Data — Since precise definitions of damage mechanisms and their associated instability conditions have not been devised for crack behavior in plane-stress and transitional stress states, only general constraints can be suggested for screening data. To assure that crack damage or fracture instability occurs under predominantly linear elastic conditions the basic criterion is that net section stress must be less than 80 percent of tensile yield strength, TYS, actually representative of that material. Additional criteria may be imposed by stress and boundary constraints characteristic to specific specimen configurations.

Middle Tension Panels — To maintain consistency with the Damage Tolerant Design Handbook [Reference 9.6.3.2.2], a related damage tolerance data document for Air Force contractors, a singular criterion,

$\mathrm{f}_{\mathrm{c}} \leq 0.8$ (TYS) $(1-2 \mathrm{a} / \mathrm{W})$ [9.6.3.2.2]

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9.6.4

CREEP AND CREEP-RUPTURE DATA — Creep is defined as time-dependent deformation of a material under an applied load. It is usually regarded as an elevated temperature phenomenon, although some materials creep at room temperature. If permitted to continue indefinitely, creep terminates in rupture. (First stage or logarithmic creep exhibited by many materials at lower temperatures is not the subject of this section.) Creep in service usually occurs under varying conditions of temperature and complex (multiaxial) stress, leading to an infinite number of stress-temperature-time combinations. Creep data for use in general design are usually obtained under conditions of constant uniform temperature and uniaxial stress. This type of data is the subject of this section.

9.6.4.1

Data Collection and Interpretation — After a desired group of creep and/or creep-rupture data have been experimentally developed or isolated in preproduction files, it is necessary to carefully collect and interpret these data in accordance with the following guidelines:

State-of-the-art for interpreting these types of creep and rupture data requires that a certain amount of judgment be allowed. The general approach will be to optimize one of several empirical equations that best follows the trend of data, using life (or time) as the dependent variable. Independent variables will include stress and temperature for rupture and isostrain creep curves, and will also include strain for isostrain creep curves.

Rupture ductility can be an exception to the above because of complex behavior and data scatter. At least a cautionary note should be given in the introductory material on times and temperatures included in rupture data. Some materials exhibit such low elongation in certain time-temperature regions that normal, reasonable values of design creep strain cannot be achieved without risk of fracture.

Interpretation of creep and rupture data should also include variables that are reflected in background data reporting requirements (discussed in the next subsection). Depending on the information content of the data, and the type of variable, it may be desirable to develop a series of equations, or to include additional physical variables in the regression analysis. The proposal should demonstrate that these additional variables have been evaluated and appropriately treated in the analysis.

The individual interpreting the data should also take note of the following special types of data, and consider the following recommendations on their use:

Specification Data—Virtually all alloys used for high-temperature applications are controlled and purchased by a process control variable generally called “spec point”. Therefore, there will often be large quantities of data available from quality control data records at the specification condition.

Data

will contain many heats, and serve as an excellent measurement source of scatter. Therefore, in

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9.6.4.2

Analysis of Data — After an acceptable data collection has been obtained and interpreted, it is possible to proceed in analyzing those data and developing mathematical models of creep and creep-rupture behavior. The objective of the procedures described in the following paragraphs is to calculate creep and rupture life as a function of test conditions and other significant variables. This calculation is done to provide an average curve and a measure of expected variability about the average. The approach that is discussed involves regression analysis to optimize the fit of an equation to the data set. The following information provides guidelines in the application of regression analysis to creep and rupture data and recommends approaches to specific problems that are frequently encountered.

General-It is assumed that life or time is the dependent variable for rupture or isostrain creep equation analysis, respectively, and logarithmic transformation of the dependent variable is normally distributed.

The data set will nearly always contain a variety of stresses and temperatures. If the data set is the product of a very well-balanced test design, good results may be obtained by independently fitting each temperature. Since this type of data set is often not available, and the approach sacrifices the opportunity for interpolation, the discussion will assume that at least temperature and stress are used as independent variables.

In order to achieve good results, it may be necessary to consider other variables. Some variables are continuous physical variables that are incorporated into regression variables, e.g., section size. Other variables may occur as discrete subsets that require modifying the regression analysis (this is discussed under

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End of chapter · Next9.7 analysis procedures for structural joint properties

9.6analysis procedures for dynamic and time dependent properties

MIL-HDBK-5J, Metallic Materials and Elements for Aerospace Vehicle Structures

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9.6.1

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9.6.1.1

(1) Total specimen separation

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9.6.1.2

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9.6.1.3

Fatigue Life Models — To clarify the fatigue data trends for a specific stress or strain ratio, a linear regression model can be applied as follows:

\log \left(\mathrm{N}_{\mathrm{i}} \text { or } \mathrm{N}_{\mathrm{f}}\right)=\mathrm{A}_{1}+\mathrm{A}_{2} \log \left(\mathrm{S}_{\max } \text { or } \Delta \epsilon\right) .

[9.6.1.3(a)]

\log \mathrm{N}_{\mathrm{j}}=\mathrm{A}_{1}+\mathrm{A}_{2}\left(\mathrm{S}_{\max }-\mathrm{A}_{3}\right)

[9.6.1.3(a)]

or by the more complicated equation [Reference 9.6.1.3]

\log N_{\mathrm{f}}=A_{1}+A_{2} \log S_{\max }+A_{3} \sqrt{\log S_{\max }}+A_{4}

[9.6.1.3(c)]

These more complex forms should only be employed in instances where they are warranted based on a distinct fatigue limit at long lives and when the simpler linear model was inadequate.

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9.6.1.4

\log \mathrm{N}_{\mathrm{f}}=\mathrm{A}_{1}+\mathrm{A}_{2} \log \left(\mathrm{S}_{\mathrm{eq}}-\mathrm{A}_{4}\right)

[9.6.1.4(a)]

where

$\mathrm{S}_{\mathrm{eq}}=\left(\Delta \mathrm{S}\right)^{\mathrm{A}_{3}}\left(\mathrm{S}_{\max }\right)^{1-\mathrm{A}_{3}}$

$\mathrm{S}_{\mathrm{eq}}=\mathrm{S}_{\max }\left(1-\mathrm{R}\right)^{\mathrm{A}_{3}}$

The equivalent stress model (and the related equivalent strain model) are derived from Reference 9.6.1.4(a).

If $\mathrm{A}_{4}$ is zero or set equal to zero, Equation 9.6.1.4(a) becomes linear in $\log \mathrm{S}_{\max }$ and $\log (1-\mathrm{R})$ , and it can be written as follows:

where ( \mathrm{B}=\mathrm{A}{2} \mathrm{~A}{3} $. Thus, if$ \mathrm{A}_{4}$ is zero, then

$\mathrm{A}_{3}=\mathrm{B} / \mathrm{A}_{2}$

Strain-controlled fatigue data generated over a range of strain ratios often can be consolidated by the following equivalent strain formulation:

\log \mathrm{N}_{\mathrm{f}}=\mathrm{A}_{1}+\mathrm{A}_{2} \log \left(\epsilon_{\mathrm{eq}}-\mathrm{A}_{4}\right)

[9.6.1.4(c)]

where

$\boldsymbol{\epsilon}_{\mathrm{eq}}=\left(\Delta \boldsymbol{\epsilon}\right)^{A_{3}}\left(\mathrm{S}_{\max } / \mathrm{E}\right)^{1-A_{3}}$

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9.6.1.5

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9.6.1.6

(1) An extreme value of the random variable inherent in the data (in this case fatigue life). If this is true, the value should be retained in future analyses.
(2) An unusual result caused by a gross deviation in material or prescribed experimental procedure or an error in calculating or recording any experimental data.

\mathrm{T}_{\mathrm{i}}=\frac{\mathrm{SR}_{\mathrm{i}}}{\left(1-\mathrm{h}_{\mathrm{i}}\right)^{1 / 2}}\left[\frac{\mathrm{RMSE}}{\mathrm{RMSE}(\mathrm{i})}\right]

[9.6.1.6(a)]

The value $\mathrm{h}_{\mathrm{i}}$ is calculated using the formula

=\frac{\mathrm{X}_{\mathrm{li}}^{2}\left(\sum \mathrm{X}_{2 \mathrm{j}}^{2}\right)-2 \mathrm{X}_{\mathrm{li}} \mathrm{X}_{2 \mathrm{i}}\left(\sum \mathrm{X}_{1 \mathrm{j}} \mathrm{X}_{2 \mathrm{j}}\right)+\mathrm{X}_{2 \mathrm{j}}^{2}\left(\sum \mathrm{X}_{1 \mathrm{j}}^{2}\right)}{\left(\sum \mathrm{X}_{2}^{2}\right)\left(\sum \mathrm{X}_{1}^{2}\right)-\left(\sum \mathrm{X}_{1} \mathrm{X}_{2}^{2}\right)}

[9.6.1.6(b)]

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9.6.1.7

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9.6.1.8

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9.6.1.9

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9.6.1.10

9.6.2

$\frac{\mathrm{da}}{\mathrm{dN}}\simeq\frac{\mathrm{\Delta a}}{\mathrm{\Delta N}}$ [ 9.6.2(a) ]

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9.6.2.1

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9.6.3

9.6.3.1

9.6.3.1.1

9.6.3.2

9.6.3.2.1

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9.6.3.2.2

$\mathrm{f}_{\mathrm{c}} \leq 0.8$ (TYS) $(1-2 \mathrm{a} / \mathrm{W})$ [9.6.3.2.2]

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9.6.4

9.6.4.1

The individual interpreting the data should also take note of the following special types of data, and consider the following recommendations on their use:

Data

will contain many heats, and serve as an excellent measurement source of scatter. Therefore, in

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9.6.4.2

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End of chapter · Next9.7 analysis procedures for structural joint properties