Fatigue and Fracture Formula Sheet

This formula sheet collects first-pass relationships used in fatigue and fracture assessment. It is not a replacement for validated material data, test evidence, design codes, or fracture-control plans. Use it to structure calculations, check assumptions, and connect stress analysis results to fatigue and crack-growth decisions.

All stresses should be clearly identified as nominal, local, principal, equivalent, structural, hot-spot, or notch stress. Fatigue data must match material condition, surface condition, environment, stress ratio, loading mode, and survival probability.

Damage-tolerance quantities

Assumed initial or missed crack size:

Critical crack size:

Crack-growth life between those sizes:

Inspection interval with inspection factor F:

Residual strength condition at detectable flaw size:

These quantities should be tied to inspection method, access, probability of detection, environment, and consequence of failure. They are not generic material constants.

Cyclic stress quantities

Maximum and minimum stress in a cycle:

Stress range:

Stress amplitude:

Mean stress:

Stress ratio:

Fully reversed loading has:

Pulsating tension has:

Fatigue calculations must state the stress ratio or mean-stress convention, because fatigue strength depends strongly on it.

S-N curve representation

A common finite-life representation is Basquin’s relation:

or:

where A, B, b, and C are fitted constants from fatigue data. On a log-log plot:

Solving for life:

Use this only over the fitted region of the S-N curve. Do not extrapolate beyond the test range without a defensible basis.

Endurance limit corrections

For materials with an endurance limit, a specimen endurance limit may be corrected for component use:

where:

• S'_e is specimen endurance limit;
• k_a is surface condition factor;
• k_b is size factor;
• k_c is loading factor;
• k_d is temperature factor;
• k_e is reliability factor;
• k_f represents miscellaneous effects.

Different design methods define these factors differently. State the source and assumptions. Many materials, including many aluminium alloys, do not have a true endurance limit.

Stress concentration and fatigue notch factor

Theoretical stress concentration factor:

Fatigue stress concentration factor:

where q is notch sensitivity. Local alternating stress:

For static yielding, local plasticity may reduce the elastic peak. For fatigue crack initiation, notches often remain damaging even when the nominal stress looks low.

Goodman mean-stress correction

Modified Goodman screening relation:

where:

• S_a is alternating stress amplitude;
• S_m is mean stress;
• S_e is endurance limit or finite-life fatigue strength;
• \sigma_{UTS} is ultimate tensile strength;
• N_f is the fatigue design factor.

Equivalent allowable alternating stress:

This relation is usually conservative for tensile mean stress in many ductile metals, but it is not a universal fatigue model.

Static checks with fatigue

Peak stress still needs static checking:

Yield screening:

Ultimate screening:

For multiaxial stress states, use a suitable equivalent stress or failure criterion. Do not assume a fatigue-safe design is automatically safe against peak yielding.

Miner cumulative damage

For variable amplitude loading:

where:

• n_i is cycles applied at stress level i;
• N_i is cycles to failure at stress level i from the S-N curve.

Failure is commonly screened at:

In design, allowable damage may be set below 1 depending on uncertainty, consequence, inspection, and method. Miner damage ignores sequence effects and should be used with care for spectra containing overloads, corrosion, crack closure, or plasticity.

Mini example: Miner damage

For two stress blocks:

and:

the cumulative damage is:

If the spectrum repeats once per year, the linear damage screen suggests:

Use this only as a screen. It assumes the S-N data, local stresses, sequence, surface condition, environment, and inspection basis remain valid.

Stress intensity factor

A common linear elastic fracture mechanics form is:

where:

• K is stress intensity factor;
• Y is geometry factor;
• \sigma is nominal stress;
• a is crack size.

Fracture occurs when:

For valid plane-strain conditions:

Fracture toughness has units such as:

Use a geometry factor and crack definition that match the actual flaw and component.

Critical crack size

Solving the stress intensity relation for critical crack size:

This gives the crack size at which fracture is expected for a given stress and geometry factor. A damage-tolerant design compares a_c with detectable flaw size, expected initial flaw size, crack-growth rate, inspection interval, and residual strength requirements.

Stress intensity range

Maximum and minimum stress intensity:

Stress intensity range:

For constant geometry factor:

Stress ratio in fracture mechanics:

when geometry does not change during the cycle.

Paris-Erdogan crack growth

Stable fatigue crack growth is often estimated with:

where C and m are material, environment, temperature, and stress-ratio dependent constants.

If Y is treated as constant and:

then:

The crack-growth life from initial crack size a_i to critical crack size a_c is:

For real components, Y may vary with crack size, and the load spectrum may not be constant. Numerical integration is often required.

Fracture toughness and plasticity

Linear elastic fracture mechanics is most appropriate when crack-tip plasticity is small relative to crack size and remaining ligament. When plasticity is significant, elastic-plastic fracture methods may use the J-integral:

A common elastic relation under plane strain is:

under plane stress:

Use J-based methods only with material data and validity checks appropriate to ductile tearing and specimen geometry.

Inspection interval concept

A simple damage-tolerance interval compares crack-growth life to inspection schedule:

where a_i is the largest crack that may be missed or the assumed initial flaw, and a_c is critical crack size. A conservative inspection interval may be chosen as a fraction of this growth life:

where F is an inspection or scatter factor selected from the design basis. This formula is conceptual; real inspection planning must include probability of detection, access, crack orientation, environment, load spectrum, and consequence of failure.

Mini example: Goodman screening

A steel component has:

The local cyclic stress has:

Then:

Goodman utilization without a design factor is:

The point is below the Goodman line, but the margin is not the whole design. A real review would also check stress concentration source, surface finish, material data, peak yielding, corrosion, variable amplitude loading, inspection, and consequence of failure.

Mini example: critical crack size

Suppose:

Critical crack size:

or about 15.8 mm. This value depends directly on geometry factor, stress, and toughness. It should not be used without checking flaw shape, section thickness, plasticity, residual stress, environment, and inspection capability.

Validation notes

For a fatigue or fracture calculation record, state:

• material condition, heat treatment, product form, surface condition, and environment;
• whether stress is nominal, local, structural, hot-spot, or notch stress;
• stress ratio, mean-stress correction, cycle count, and load-spectrum source;
• S-N curve, endurance correction factors, or crack-growth constants used;
• assumed initial flaw size, detectable flaw size, critical crack size, and inspection interval;
• NDE method, access limitation, and detection capability used in the damage-tolerance argument;
• acceptance criteria for damage, crack growth, residual strength, and repair.

The formulas are only useful when they are traceable to the real detail, real spectrum, and real inspection capability.

Common cautions

Do not use tensile strength as a substitute for fracture toughness. Do not apply a polished-specimen S-N curve directly to a real part with notches, welds, corrosion, or rough surfaces. Do not treat the Miner rule as exact. Do not assume an endurance limit exists for every material. Do not use Paris-law constants outside their fitted range. Do not plan inspection intervals without considering probability of detection and critical crack size.