Predicting Service Life Under Cyclic Stresses

Components made from metal can withstand impressive static forces, yet repeatedly applying even modest stresses often leads to sudden failure. This insidious form of damage, known as fatigue, arises because microscopic cracks initiate and grow each time the material flexes. The seemingly solid piece behaves like a living object that "remembers" past loading history. Engineers therefore devote considerable effort to estimating how many cycles a part can safely endure so maintenance or replacement can occur before fracture. The calculator on this page assists with such assessments by combining two widely used models: Basquin’s empirical S–N relationship, which predicts life under a single constant-amplitude stress, and Miner's linear damage accumulation rule that extends the prediction to variable loading. The goal is not to replace detailed laboratory testing but to provide a first approximation that highlights trends and encourages a deeper understanding of fatigue phenomena.

Fatigue is governed by complex microstructural processes. At high stress levels near a material’s yield strength, slip bands form and merge to create cracks. At lower stresses, inclusions or surface imperfections serve as crack nuclei. Once a crack forms, each subsequent cycle extends it incrementally. Eventually the remaining cross-section can no longer support the applied load and the part fractures abruptly, often with little prior warning. Factors such as surface finish, residual stresses, temperature, corrosion, and loading frequency influence the rate at which damage accumulates. To make the problem tractable, engineers represent the aggregate effect of these influences through experimentally measured S–N curves that relate stress amplitude to the number of cycles to failure.

Basquin’s Equation and the S–N Curve

When plotted on a log–log scale, the relationship between stress amplitude and fatigue life for many metals approximates a straight line over the high-cycle regime. This observation led to Basquin’s equation:

Here $\sigma a_{}$ is the stress amplitude of a fully reversed loading cycle, $\sigma {\prime }^f$ is the fatigue strength coefficient, $N$ is the number of cycles to failure, and $b$ is the fatigue exponent, typically a negative value between −0.05 and −0.15 for metals. Rearranging gives the life prediction:

The equation captures the intuitive idea that a higher stress amplitude dramatically shortens life. A component designed for a lower allowable stress will endure orders of magnitude more cycles. In practice, engineers extract the constants $\sigma {\prime }^f$ and $b$ from fatigue tests performed on specimens of the same material and surface condition as the component of interest. If direct data are unavailable, typical values from literature may be used cautiously.

Miner’s Rule for Variable Amplitude Loading

Real structures rarely experience a single stress amplitude. Instead, loads vary over time. Suppose a part endures $n$₁ cycles at stress $\sigma$_a1, then $n$₂ cycles at $\sigma$_a2, and so on. Each block of cycles consumes a fraction of the component’s total life at the respective stress level. Miner's rule estimates the cumulative damage $D$ as the sum of the individual ratios of applied cycles to allowable cycles:

Failure is predicted when the accumulated damage reaches unity ($D=1$). Although simplistic, Miner's rule offers a linear and easily computable way to handle load spectra. The calculator implements this logic for up to three distinct stress levels. Users may leave unused fields blank; the script ignores those cases.

Interpreting the Results

After entering the material parameters and loading history, the calculator reports two quantities: the cumulative damage and the equivalent life in cycles. The damage fraction reveals how close the part is to the failure threshold. For example, a result of $D=0.3$ indicates roughly thirty percent of life consumed, suggesting ample margin if conditions remain similar. Conversely, a damage value exceeding one implies that the component has already surpassed its expected endurance. The equivalent life expresses the total number of cycles the part could have survived under the actual mix of loads. By comparing this value with planned service cycles, engineers can schedule inspections or replacements responsibly.

Typical Material Parameters

The constants used in Basquin’s equation vary widely across alloys and heat treatments. The table lists representative values for several metals subjected to fully reversed bending, providing a starting point for preliminary calculations.

Material	σ′f (MPa)	b
Low-carbon steel	1000	-0.09
Aluminum alloy 6061-T6	900	-0.10
Titanium alloy Ti-6Al-4V	1100	-0.08
Magnesium alloy AZ91D	700	-0.11

These coefficients are approximate; authentic design should reference material-specific fatigue data and apply correction factors for surface finish, size, temperature, and reliability. Nonetheless, the numbers illustrate how materials with similar static strengths may exhibit markedly different fatigue behavior.

Limitations and Practical Considerations

While Basquin’s and Miner's formulations are prevalent due to their simplicity, they embody assumptions that limit accuracy. Basquin's equation presumes high-cycle elastic behavior; it underestimates damage when stresses approach the yield strength where plasticity occurs. Miner's rule neglects load order effects and interactions between different stress levels. For example, overload cycles may retard subsequent crack growth, while small cycles may do negligible damage until a crack exceeds a threshold size. The method also assumes a clear definition of stress amplitude, yet real components experience multiaxial states and mean stresses that shift the S–N curve. Engineers often incorporate safety factors or more sophisticated approaches such as strain-life (ε–N) methods, fracture mechanics, or spectrum loading analyses to capture these nuances.

Despite these caveats, simple fatigue estimates remain invaluable during the conceptual design stage. They help identify weak points, compare alternative materials, and communicate with stakeholders about maintenance intervals. The visualization of damage accumulation encourages asset managers to collect accurate load histories and monitor critical components. By coupling empirical S–N data with Miner's rule, the calculator provides a transparent window into the otherwise invisible process of fatigue. It reminds us that even stationary structures, from bridges to aircraft wings, are subject to a countdown dictated by the stresses we impose upon them. Thoughtful engineering involves listening to that countdown and planning accordingly.