Bathtub Curve

The early-life period of a component or system characterized by a higher-than-normal failure rate that decreases over time (Weibull beta < 1). Infant mortality failures are caused by latent defects introduced during manufacturing, installation, or commissioning — they are not age-related degradation but rather quality problems that manifest early. Common causes include material defects, welding flaws, installation errors, contamination, and incorrect settings.

The late-life period of a component or system characterized by an increasing failure rate over time (Weibull beta > 1), caused by progressive degradation mechanisms such as fatigue, corrosion, erosion, creep, and material aging. Wear-out failures are predictable and age-related, making them amenable to time-based or condition-based preventive maintenance strategies.

A failure that occurs with approximately constant probability per unit time, independent of the item's age or accumulated operating time (Weibull beta approximately 1). Random failures are caused by unpredictable combinations of stress and strength, external events, or operating conditions that cannot be anticipated by age-based inspection or maintenance. They are equally likely to occur at any point during the useful life period.

The instantaneous rate of failure at time t, given that the item has survived to time t. Hazard rate (also called the instantaneous failure rate or hazard function) describes how the risk of failure changes over the item's life. Unlike failure rate (which is an average), hazard rate is a time-dependent function that captures the changing nature of failure risk as equipment ages.

A continuous probability distribution widely used in reliability engineering to model the time-to-failure of components and systems. The two-parameter Weibull distribution is defined by the shape parameter (beta) and scale parameter (eta). Its versatility lies in its ability to model increasing, constant, or decreasing failure rates depending on the shape parameter value, making it suitable for all phases of the bathtub curve.

The Weibull shape parameter (beta, also written as b or k) that determines the form of the failure distribution and the nature of the failure rate over time. Beta < 1 indicates decreasing failure rate (infant mortality); beta = 1 indicates constant failure rate (random failures, equivalent to exponential distribution); beta > 1 indicates increasing failure rate (wear-out). The value of beta reveals the physics of the dominant failure mechanism.