1.4basic principles

1.4.1.1

Basis — Primary static design properties are provided for the following conditions:

Tension $F_{tu}$ and $F_{ty}$ Compression $F_{cy}$ Shear $F_{su}$ Bearing $F_{bru}$ and $F_{bry}$

These design properties are presented as A- and B- or S-basis room temperature values for each alloy. Design properties for other temperatures, when determined in accordance with Section 1.4.1.3, are regarded as having the same basis as the corresponding room temperature values.

Elongation and reduction of area design properties listed in room temperature property tables represent procurement specification minimum requirements, and are designated as S-values. Elongation and reduction of area at other temperatures, as well as moduli, physical properties, creep properties, fatigue properties and fracture toughness properties are all typical values unless another basis is specifically indicated.

Use of B-Values — The use of B-basis design properties is permitted in design by the Air Force, the Army, the Navy, and the Federal Aviation Administration, subject to certain limitations specified by each agency. Reference should be made to specific requirements of the applicable agency before using B-values in design.

1.4.1.2

Statistically Calculated Values — Statistically calculated values are S (since 1975), T99 and T90. S, the minimum properties guaranteed in the material specification, are calculated using the same requirements and procedure as AMS and is explained in Chapter 9. T99 and T90 are the local tolerance bounds, and are defined and may be computed using the data requirements and statistical procedures explained in Chapter 9.

1.4.1.3

Ratioed Values — A ratioed design property is one that is determined through its relationship with an established design value. This may be a tensile stress in a different grain direction from the established design property grain direction, or it may be another stress property, e.g., compression, shear or bearing. It may also be the same stress property at a different temperature. Refer to Chapter 9 for specific data requirements and data analysis procedures.

Derived properties are presented in two manners. Room temperature derived properties are presented in tabular form with their baseline design properties. Other than room temperature derived properties are presented in graphical form as percentages of the room temperature value. Percentage

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1.4.3.1

Poisson's Ratio Effect — A normal strain is that which is associated with a normal stress; a normal strain occurs in the direction in which its associated normal stress acts. Normal strains that result from an increase in length are designated as positive (+) and those that result in a decrease in length are designated as negative (-).

Under the condition of uniaxial loading, strain varies directly with stress. The ratio of stress to strain has a constant value (E) within the elastic range of the material, but decreases when the proportional limit is exceeded (plastic range). Axial strain is always accompanied by lateral strains of opposite sign in the two directions mutually perpendicular to the axial strain. Under these conditions, the absolute value of a ratio of lateral strain to axial strain is defined as Poisson’s ratio. For stresses within the elastic range, this ratio is approximately constant. For stresses exceeding the proportional limit, this ratio is a function of the axial strain and is then referred to as the lateral contraction ratio. Information on the variation of Poisson’s ratio with strain and with testing direction is available in Reference 1.4.3.1.

Under multiaxial loading conditions, strains resulting from the application of each directional load are additive. Strains must be calculated for each of the principal directions taking into account each of the principal stresses and Poisson’s ratio (see Equation 1.3.7 for biaxial loading).

1.4.3.2

Shear Strain — When an element of uniform thickness is subjected to pure shear, each side of the element will be displaced in opposite directions. Shear strain is computed by dividing this total displacement by the right angle distance separating the two sides.

1.4.3.3

Strain Rate — Strain rate is a function of loading rate. Test results are dependent upon strain rate, and the ASTM testing procedures specify appropriate strain rates. Design properties in this Handbook were developed from test data obtained from coupons tested at the stated strain rate or up to a value of 0.01 in./in./min, the standard maximum static rate for tensile testing materials per specification ASTM E 8.

1.4.3.4

Elongation and Reduction of Area — Elongation and reduction of area are measured in accordance with specification ASTM E 8.

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1.4.4.1

Modulus of Elasticity (E) — Referring to Figure 1.4.4, it is noted that the initial part of stress-strain curves are straight lines. This indicates a constant ratio between stress and strain. Numerical values of such ratios are defined as the modulus of elasticity, and denoted by the letter E. This value applies up to the proportional limit stress at which point the initial slope of the stress-strain curve then decreases. Modulus of elasticity has the same units as stress. See Equation 1.3.4 (b).

Other moduli of design importance are tangent modulus, $E_{\mathrm{t}}$, and secant modulus, $E_{\mathrm{s}}$. Both of these moduli are functions of strain. Tangent modulus is the instantaneous slope of the stress-strain curve at any selected value of strain. Secant modulus is defined as the ratio of total stress to total strain at any selected value of strain. Both of these moduli are used in structural element designs. Except for materials such as those described with discontinuous behaviors, such as the upper stress-strain curve in Figure 1.4.4, tangent modulus is the lowest value of modulus at any state of strain beyond the proportional limit. Similarly, secant modulus is the highest value of modulus beyond the proportional limit.

Clad aluminum alloys may have two separate modulus of elasticity values, as indicated in the typical stress-strain curve shown in Figure 1.4.4. The initial slope, or primary modulus, denotes a response of both the low-strength cladding and higher-strength core elastic behaviors. This value applies only up to the proportional limit of the cladding. For example, the primary modulus of 2024-T3 clad sheet applies only up to about 6 ksi. Similarly, the primary modulus of 7075-T6 clad sheet applies only up to approximately 12 ksi. A typical use of primary moduli is for low amplitude, high frequency fatigue. Primary moduli are not applicable at higher stress levels. Above the proportional limits of cladding materials, a short transition range occurs while the cladding is developing plastic behavior. The material then exhibits a secondary elastic modulus up to the proportional limit of the core material. This secondary modulus is the slope of the second straight line portion of the stress-strain curve. In some cases, the cladding is so little different from the core material that a single elastic modulus value is used.

1.4.4.2

Tensile Proportional Limit Stress (Fip) — The tensile proportional limit is the maximum stress for which strain remains proportional to stress. Since it is practically impossible to determine precisely this point on a stress-strain curve, it is customary to assign a small value of plastic strain to identify the corresponding stress as the proportional limit. In this Handbook, the tension and compression proportional limit stress corresponds to a plastic strain of 0.0001 in./in.

1.4.4.3

Tensile Yield Stress (TYS or $F_{y}$ ) — Stress-strain diagrams for some ferrous alloys exhibit a sharp break at a stress below the tensile ultimate strength. At this critical stress, the material elongates considerably with no apparent change in stress. See the upper stress-strain curve in Figure 1.4.4. The stress at which this occurs is referred to as the yield point. Most nonferrous metallic alloys and most high strength steels do not exhibit this sharp break, but yield in a monotonic manner. This condition is also illustrated in Figure 1.4.4. Permanent deformation may be detrimental, and the industry adopted 0.002 in./in. plastic strain as an arbitrary limit that is considered acceptable by all regulatory agencies. For tension and compression, the corresponding stress at this offset strain is defined as the yield stress (see Figure 1.4.4). This value of plastic axial strain is 0.002 in./in. and the corresponding stress is defined as the yield stress. For practical purposes, yield stress can be determined from a stress-strain diagram by extending a line parallel to the elastic modulus line and offset from the origin by an amount of 0.002 in./in.

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1.4.4.4

Tensile Ultimate Stress (TUS or $F_{\mathrm{ty}}$ ) — Figure 1.4.4 shows how the tensile ultimate stress is determined from a stress-strain diagram. It is simply the maximum stress attained. It should be noted that all stresses are based on the original cross-sectional dimensions of a test specimen, without regard to the lateral contraction due to Poisson's ratio effects. That is, all strains used herein are termed engineering strains as opposed to true strains which take into account actual cross sectional dimensions. Ultimate tensile stress is commonly used as a criterion of the strength of the material for structural design, but it should be recognized that other strength properties may often be more important.

1.4.4.5

Elongation (e) — An additional property that is determined from tensile tests is elongation. This is a measure of ductility. Elongation, also stated as total elongation, is defined as the permanent increase in gage length, measured after fracture of a tensile specimen. It is commonly expressed as a percentage of the original gage length. Elongation is usually measured over a gage length of 2 inches for rectangular tensile test specimens and in 4D (inches) for round test specimens. Welded test specimens are exceptions. Refer to the applicable material specification for applicable specified gage lengths. Although elongation is widely used as an indicator of ductility, this property can be significantly affected by testing variables, such as thickness, strain rate, and gage length of test specimens. See Section 1.4.1.1 for data basis.

1.4.4.6

Reduction of Area (RA) — Another property determined from tensile tests is reduction of area, which is also a measure of ductility. Reduction of area is the difference, expressed as a percentage of the original cross sectional area, between the original cross section and the minimum cross sectional area adjacent to the fracture zone of a tested specimen. This property is less affected by testing variables than elongation, but is more difficult to compute on thin section test specimens. See Section 1.4.1.1 for data basis.

1.4.5.1

Compressive Ultimate Stress ( $F_{c u}$ ) – Since the actual failure mode for the highest tension and compression stress is shear, the maximum compression stress is limited to $F_{t u}$. The driver for all the analysis of all structure loaded in compression is the slope of the compression stress strain curve, the tangent modulus.

1.4.5.2

Compressive Yield Stress (CYS or $F_{c y}$ ) — Compressive yield stress is measured in a manner identical to that done for tensile yield strength. It is defined as the stress corresponding to 0.002 in./in. plastic strain.

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1.4.6.1

Modulus of Rigidity (G)—This property is the initial slope of the shear stress-strain curve. It is also referred to as the modulus of elasticity in shear. The relation between this property and the modulus of elasticity in tension is expressed for homogeneous isotropic materials by the following equation:

1.4.6.2

Proportional Limit Stress in Shear (Fsp) — This property is of particular interest in connection with formulas which are based on considerations of linear elasticity, as it represents the limiting value of shear stress for which such formulas are applicable. This property cannot be determined directly from torsion tests.

1.4.6.3

Yield and Ultimate Stresses in Shear (SYS or Fsy) and (SUS or Fsu) — These properties, as usually obtained from ASTM test procedures tests, are not strictly basic properties, as they will depend on the shape of the test specimen. In such cases, they should be treated as moduli and should not be combined with the same properties obtained from other specimen configuration tests.

Design values reported for shear ultimate stress (Fsu) in room temperature property tables for aluminum and magnesium thin sheet alloys are based on “punch” shear type tests except when noted. Heavy section test data are based on “pin” tests. Thin aluminum products may be tested to ASTM B 831, which is a slotted shear test. Thicker aluminums use ASTM B 769, otherwise known as the Amsler shear test. These two tests only provide ultimate strength. Shear data for other alloys are obtained from pin tests, except where product thicknesses are insufficient. These tests are used for other alloys; however, the standards don’t specifically cover materials other than aluminum

1.4.7.1

Bearing Yield and Ultimate Stresses (BYS or $F_{b y}$ ) and (BUS or $F_{b u}$ ) —BUS is the maximum stress withstood by a bearing specimen. BYS is computed from a bearing stressdeformation curve by drawing a line parallel to the initial slope at an offset of 0.02 times the pin diameter.

Tabulated design properties for bearing yield stress (F_{bry}) and bearing ultimate stress (F_{bru}) are provided throughout the Handbook for edge margins of e/D = 1.5 and 2.0. Bearing values for e/D of 1.5 are not intended for designs of e/D < 1.5. Bearing values for e/D < 1.5 must be substantiated by adequate

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1.4.8.1

Low Temperature — Temperatures below room temperature generally cause an increase in strength properties of metallic alloys. Ductility, fracture toughness, and elongation usually decrease. For specific information, see the applicable chapter and references noted therein.

1.4.8.2

Elevated Temperature — Temperatures above room temperature usually cause a decrease in the strength properties of metallic alloys. This decrease is dependent on many factors, such as temperature and the time of exposure which may degrade the heat treatment condition, or cause a metallurgical change. Ductility may increase or decrease with increasing temperature depending on the same variables. Because of this dependence of strength and ductility at elevated temperatures on many variables, it is emphasized that the elevated temperature properties obtained from this Handbook be applied for only those conditions of exposure stated herein.

The effect of temperature on static mechanical properties is shown by a series of graphs of property (as percentages of the room temperature allowable property) versus temperature. Data used to construct these graphs were obtained from tests conducted over a limited range of strain rates. Caution should be exercised in using these static property curves at very high temperatures, particularly if the strain rate intended in design is much less than that stated with the graphs. The reason for this concern is that at very low strain rates or under sustained loads, plastic deformation or creep deformation may occur to the detriment of the intended structural use.

1.4.8.2.1

Creep and Stress-Rupture Properties — Creep is defined as a time-dependent deformation of a material while 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. Since creep in service is usually typified by complex conditions of loading and temperature, the number of possible stress-temperature-time profiles is infinite. For economic reasons, creep data for general design use are usually obtained under conditions of constant uniaxial loading and constant temperature in accordance with Reference 1.4.8.2.1(a). Creep data are sometimes obtained under conditions of cyclic uniaxial loading and constant temperature, or constant uniaxial loading and variable temperatures. Section 9.3.6 provides a limited amount of creep data analysis procedures. It is recognized that, when significant creep appears likely to occur, it may be necessary to test under simulated service conditions because of difficulties posed in attempting to extrapolate from simple to complex stress-temperature-time conditions.

Creep damage is cumulative similar to plastic strain resulting from multiple static loadings. This damage may involve significant effects on the temper of heat treated materials, including annealing, and

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1.4.8.2.2

Creep-Rupture Curve — Results of tests conducted under constant loading and constant temperature are usually plotted as strain versus time up to rupture. A typical plot of this nature is shown in Figure 1.4.8.2.2. Strain includes both the instantaneous deformation due to load application and the plastic strain due to creep. Other definitions and terminology are provided in Section 9.3.6.2.

Figure description:

Typical Creep Curve Showing Three Stages of Creep

Creep Curve

Annotations

• Initial Stage (region - position: First interval on Time axis)
• Second Stage (region - position: Middle interval on Time axis)
• Third Stage (region - position: Final interval on Time axis)
• Creep Intercept (text_label - position: y-axis intercept of the dashed extrapolation line)
• Slope = minimum creep rate (text_label - position: Slope of the Second Stage linear region)
• Transition Point (text_label - position: Boundary between Second and Third Stages)
• Rupture Point (text_label - position: Terminal point of the Third Stage curve)

Figure 1.4.8.2.2. Typical creep-rupture curve.

1.4.8.2.3

Creep or Stress-Rupture Presentations — Results of creep or stress-rupture tests conducted over a range of stresses and temperatures are presented as curves of stress versus the logarithm of time to rupture. Each curve represents an average, best-fit description of measured behavior. Modification of such curves into design use are the responsibility of the design community since material applications and regulatory requirements may differ. Refer to Section 9.3.6 for data reduction and presentation methods and References 1.4.8.2.1(b) and (c).

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1.4.9.1

Terminology — A number of symbols and definitions are commonly used to describe fatigue test conditions, test results and data analysis techniques. The most important of these are described in Section 9.3.4.2.

1.4.9.2

Graphical Display of Fatigue Data — Results of axial fatigue tests are reported on S-N and ε - N diagrams. Figure 1.4.9.2(a) shows a family of axial load S-N curves. Data for each curve represents a separate R-value.

S-N and $\varepsilon$-N diagrams are shown in this Handbook with the raw test data plotted for each stress or strain ratio or, in some cases, for a single value of mean stress. A best-fit curve is drawn through the data at each condition. Rationale used to develop best-fit curves and the characterization of all such curves in a single diagram is explained in Section 9.3.4. For load control test data, individual curves are usually based on an equivalent stress that consolidates data for all stress ratios into a single curve. Refer to Figure 1.4.9.2(b). For strain control test data, an equivalent strain consolidation method is used.

Elevated temperature fatigue test data are treated in the same manner as room temperature data, as long as creep is not a significant factor and room temperature analysis methods can be applied. In the limited number of cases where creep strain data have been recorded as a part of an elevated temperature fatigue test series, S-N (or ε - N) plots are constructed for specific creep strain levels. This is provided in addition to the customary plot of maximum stress (or strain) versus cycles to failure.

The above information may not apply directly to the design of structures for several reasons. First, Handbook information may not take into account specific stress concentrations unique to any given structural design. Design considerations usually include stress concentrations caused by re-entrant corners, notches, holes, joints, rough surfaces, structural damage, and other conditions. Localized high stresses induced during the fabrication of some parts have a much greater influence on fatigue properties than on static properties. These factors significantly reduce fatigue life below that which is predictable by estimating smooth specimen fatigue performance with estimated stresses due to fabrication. Fabricated parts have been found to fail at less than 50,000 cycles of loading when the nominal stress was far below that which could be repeated many millions of times using a smooth-machined test specimen.

Notched fatigue specimen test data are shown in various Handbook figures to provide an understanding of deleterious effects relative to results for smooth specimens. All of the mean fatigue curves published in this Handbook, including both the notched fatigue and smooth specimen fatigue curves, require modification into allowables for design use. Such factors may impose a penalty on cyclic life or upon stress. This is a responsibility for the design community. Specific reductions vary between users of such information, and depending on the criticality of application, sources of uncertainty in the analysis, and requirements of the certificating activity. References 1.4.9.2(a) and (b) contain more specific information on fatigue testing procedures, organization of test results, influences of various factors, and design considerations.

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1.4.11.1

Biaxial Modulus of Elasticity — Referring to Figure 1.4.11, it is noted that the original portion of each stress-strain curve is essentially a straight line. In uniaxial tension or compression, the slope of this line is defined as the modulus of elasticity. Under biaxial loading conditions, the initial slope of such curves is defined as the biaxial modulus. It is a function of biaxial stress ratio and Poisson’s ratio. See Equation 1.3.7.4.

1.4.11.2

Biaxial Yield Stress — Biaxial yield stress is defined as the maximum principal stress corresponding to 0.002 in./in. plastic strain in the same direction, as determined from a test curve.

In the design of aerospace structures, biaxial stress ratios other than those normally used in biaxial testing are frequently encountered. Information can be combined into a single diagram to enable interpolations at intermediate biaxial stress ratios, as shown in Figure 1.4.11.2. An envelope is constructed through test results for each tested condition of biaxial stress ratios. In this case, a typical biaxial yield stress envelope is identified. In the preparation of such envelopes, data are first reduced to nondimensional form (percent of uniaxial tensile yield stress in the specified reference direction), then a best-fit curve is fitted through the nondimensionalized data. Biaxial yield strength allowables are then obtained by multiplying the uniaxial $\mathrm{F}_{\mathrm{ty}}$ (or $\mathrm{F}_{\mathrm{cy}}$ ) allowable by the applicable coordinate of the biaxial stress ratio curve. To avoid possible confusion, the reference direction used for the uniaxial yield strength is indicated on each figure.

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1.4.11.3

Biaxial Ultimate Stress — Biaxial ultimate stress is defined as the highest nominal principal stress attained in specimens of a given configuration, tested at a given biaxial stress ratio. This property is highly dependent upon geometric configuration of the test parts. Therefore, such data should be limited in use to the same design configurations.

The method of presenting biaxial ultimate strength data is similar to that described in the preceding section for biaxial yield strength. Both biaxial ultimate strength and corresponding uniform elongation data are reported, when available, as a function of biaxial stress ratio test conditions.

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1.4.12.1

Brittle Fracture — For materials that have little capacity for plastic flow, or for flaw and structural configurations, which induce triaxial tension stress states adjacent to the flaw, component behavior is essentially elastic until the fracture stress is reached. Then, a crack propagates from the flaw suddenly and completely through the component. A convenient illustration of brittle fracture is a typical load-compliance record of a brittle structural component containing a flaw, as illustrated in Figure 1.4.12.1. Since little or no plastic effects are noted, this mode is termed brittle fracture.

This mode of fracture is characteristic of the very high-strength metallic materials under plane-strain conditions.

Figure description:

Structural Load vs. Deformation Chart

Structural Load vs. Deformation

Annotations

• Fracture at Maximum Load (text_label - position: at point ($\Delta_{max}, P_{max}$))

Figure 1.4.12.1. Typical load-deformation record of a structural component containing a flaw subject to brittle fracture.

1.4.12.2

Brittle Fracture Analysis — The application of linear elastic fracture mechanics has led to the stress intensity concept to relate flaw size, component geometry, and fracture toughness. In its very general form, the stress intensity factor, K, can be expressed as

$\mathrm{K}=\mathrm{f}\sqrt{\mathrm{a}}\mathrm{Y},\mathrm{k}\mathrm{s}\mathrm{i}\cdot\mathrm{i}\mathrm{n}.^{1/2}$ [1.4.12.2]

where

f = stress applied to the gross section, ksi

• a = measure of flaw size, inches

Y = factor relating component geometry and flaw size, nondimensional. See Reference 1.4.12.2(a) for values.

For every structural material, which exhibits brittle fracture (by nature of low ductility or plane-strain stress conditions), there is a lower limiting value of K termed the plane-strain fracture toughness, $\mathrm{K}_{\mathrm{lc}}$.

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1.4.12.3

Critical Plane-Strain Fracture Toughness — A tabulation of fracture toughness data is printed in the general discussion prefacing most alloy chapters in this Handbook. These critical plane-strain fracture toughness values have been determined in accordance with recommended ASTM testing practices. This information is provided for information purposes only due to limitations in available data quantities and product form coverages. The statistical reliability of these properties is not known. Listed properties generally represent the average value of a series of test results.

Fracture toughness of a material commonly varies with grain direction. When identifying either test results or a general critical plane strain fracture toughness average value, it is customary to specify specimen and crack orientations by an ordered pair of grain direction symbols per ASTM E399. [Reference 1.4.12.2(a).] The first digit denotes the grain direction normal to the crack plane. The second digit denotes the grain direction parallel to the fracture plane. For flat sections of various products, e.g., plate, extrusions, forgings, etc., in which the three grain directions are designated (L) longitudinal, (T) transverse, and (S) short transverse, the six principal fracture path directions are: L-T, L-S, T-L, T-S, S-L and S-T. Figure 1.4.12.3(a) identifies these orientations. For cylindrical sections where the direction of principle deformation is parallel to the longitudinal axis of the cylinder, the reference directions are identified as in Figure 1.4.12.3(b), which gives examples for a drawn bar. The same system would be useful for extrusions or forged parts having circular cross section.

Figure description:

Key Entities & Information:

• Grain Directions (Designated for a flat material section:

  • L (Length: Longitudinal, Rolling Direction, Extrusion Direction, or Axis of Forging.
  • T (Width: Long Transverse.
  • S (Thickness: Short Transverse.

• Principal Fracture Path Directions (Ordered Pairs:

  • First Letter: Direction normal to the crack plane.
  • Second Letter: Direction of crack propagation (parallel to the fracture plane.
  • Identified Orientations: L-T, L-S, T-L, T-S, S-L, and S-T.

• Visual Representation: 3D diagram showing six fracture toughness specimens oriented relative to the material's L, T, and S axes.

Figure 1.4.12.3(a). Typical principal fracture path directions.

Figure description:

Key Information:

• Subject: Typical principal fracture path directions for cylindrical shapes.
• Main Entities: A cylindrical volume containing six notched fracture specimens at distinct orientations.
• Specimen Orientations (Load direction - Propagation direction:
  • C-R: Circumferential-Radial
  • R-C: Radial-Circumferential
  • R-L: Radial-Longitudinal
  • C-L: Circumferential-Longitudinal
  • L-R: Longitudinal-Radial
  • L-C: Longitudinal-Circumferential

Figure 1.4.12.3(b). Typical principal fracture path directions for cylindrical shapes.

1.4.12.3.1

Environmental Effects — Cyclic loading, even well below the fracture threshold stress, may result in the propagation of flaws, leading to fracture. Strain rates in excess of standard static rates may cause variations in fracture toughness properties. There are significant influences of temperature

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1.4.12.4

Fracture in Plane-Stress and Transitional-Stress States — Plane-strain conditions do not describe the condition of certain structural configurations which are either relatively thin or exhibit appreciable ductility. In these cases, the actual stress state may approach the opposite extreme, plane-stress, or, more generally, some intermediate- or transitional-stress state. The behavior of flaws and cracks under these conditions is different from those of plane-strain. Specifically, under these conditions, significant plastic zones can develop ahead of the crack or flaw tip, and stable extension of the discontinuity occurs as a slow tearing process. This behavior is illustrated in a compliance record by a significant nonlinearity prior to fracture as shown in Figure 1.4.12.4. This nonlinearity results from the alleviation of stress at the crack tip by causing plastic deformation.

Figure description:

Load-Displacement Analysis

Load vs. Displacement Curve

Annotations

• P_0 (text_label - position: y-axis at threshold load)
• P_I (text_label - position: y-axis at maximum load)
• Threshold of nonlinear behavior (slow crack growth and plasticity (text_label - position: At intersection of P_0 and the curve (approx. Δ=25.0)))
• Maximum load and fracture (text_label - position: At the peak of the curve (approx. Δ=75.0))
• P_0 reference line (horizontal_line - position: y = 55)
• P_I reference line (horizontal_line - position: y = 96)

Figure 1.4.12.4. Typical load-deformation record for non-plane strain fracture.

1.4.12.4.1

Analysis of Plane-Stress and Transitional-Stress State Fracture — The basic concepts of linear elastic fracture mechanics as used in plane-strain fracture analysis also applies to these conditions. The stress intensity factor concept, as expressed in general form by Equation 1.4.12.2, is used to relate load or stress, flaw size, component geometry, and fracture toughness.

However, interpretation of the critical flaw dimension and corresponding stress has two possibilities. This is illustrated in Figure 1.4.12.4.1. One possibility is the onset of nonlinear displacement with increasing load. The other possibility identifies the fracture condition, usually very close to the

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1.4.12.5

Apparent Fracture Toughness Values for Plane-Stress and Transitional-Stress States — When available, each alloy chapter contains graphical formats of stress versus flaw size. This is provided for each temper, product form, grain direction, thickness, and specimen configuration. Data points shown in these graphs represent the initial flaw size and maximum stress achieved. These data have been screened to assure that an elastic instability existed at fracture, consistent with specimen type. The average $\mathrm{K}_{\mathrm{app}}$ curve, as defined in the following subsections, is shown for each set of data.

1.4.12.5.1

Middle-Tension Panels — The calculation of apparent fracture toughness for middle-tension panels is given by the following equation.

[1.4.12.5.1(a)]

Data used to compute $\mathrm{K}_{\mathrm{app}}$ values have been screened to ensure that the net section stress at failure did not exceed 80 percent of the tensile yield strength; that is, they satisfied the criterion:

$\mathrm{f}_{\mathrm{c}} \leq 0.8(\mathrm{TY} / \mathrm{S}) /(1-2 \mathrm{a} / \mathrm{W})$ [1.4.12.5.1(b)]

This criterion assures that the fracture was an elastic instability and that plastic effects are negligible.

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1.4.13.1

Fatigue Crack Growth — Fatigue crack growth is manifested as the growth or extension of a crack under cyclic loading. This process is primarily controlled by the maximum load or stress ratio. Additional factors include environment, loading frequency, temperature, and grain direction. Certain factors, such as environment and loading frequency, have interactive effects. Environment is important from a potential corrosion viewpoint. Time at stress is another important factor. Standard testing procedures are documented in Reference 1.4.13.1.

Fatigue crack growth data presented herein are based on constant amplitude tests. Crack growth behaviors based on spectrum loading cycles are beyond the scope of this Handbook. Constant amplitude data consist of crack length measurements at corresponding loading cycles. Such data are presented as crack growth curves as shown in Figure 1.4.13.1(a).

Since the crack growth curve is dependent on initial crack length and the loading conditions, the above format is not the most efficient form to present information. The instantaneous slope, $\Delta \mathrm{a} / \Delta \mathrm{N}$, corresponding to a prescribed number of loading cycles, provides a more fundamental characterization of this behavior. In general, fatigue crack growth rate behavior is evaluated as a function of the applied stress intensity factor range, $\Delta \mathrm{K}$, as shown in Figure 1.4.13.1(b).

1.4.13.2

Fatigue Crack Growth Analysis — It is known that fatigue-crack-growth behavior under constant-amplitude cyclic conditions is influenced by maximum cyclic stress, $S_{\max }$, and some measure of cyclic stress range, $\Delta S$ (such as stress ratio, R, or minimum cyclic stress, $S_{\min }$ ), the instantaneous crack size, a, and other factors such as environment, frequency, and temperature. Thus, fatigue-crack-growth rate behavior can be characterized, in general form, by the relation

[1.4.13.3(a)]

By applying concepts of linear elastic fracture mechanics, the stress and crack size parameters can be combined into the stress-intensity factor parameter, K, such that Equation 1.4.13.3(a) may be simplified

to

$\mathrm{da} / \mathrm{dN} \approx \Delta \mathrm{a} / \Delta \mathrm{N}=\mathrm{g}\left(\mathrm{K}_{\max }, \Delta \mathrm{K}, \ldots\right)$

[1.4.13.3(b)]

where

K_max = the maximum cyclic stress-intensity factor ΔK = (1-R)K_max, the range of the cyclic stress-intensity factor, for R ≥ 0 ΔK = K_max, for R ≤ 0.

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1.4.13.2

Paris, P.C., "The Fracture Mechanics Approach to Fatigue," Proc. 10th Sagamore Conference, p. 107, Syracuse University Press (1965).

1.4.13.3

Fatigue Crack Growth Data Presentation — Fatigue crack growth rate data for constant amplitude cyclic loading conditions are presented as logarithmic plots of da/dN versus AK. Such information, such as that illustrated in Figure 1.4.13.3, are arranged by material alloy and heat treatment condition. Each curve represents a specific stress ratio, R, environment, and cyclic loading frequency. Specific details regarding test procedures and data interpolations are presented in Chapter 9.

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