5 Accelerated Life Testing

5.1 Introduction

Accelerated Life Testing (ALT) (Meeker and Escobar 1998; Nelson 1990) subjects products to elevated stress levels — temperature, voltage, mechanical load, humidity — to induce failures more quickly than would occur under normal operating conditions. The acceleration relationship is then used to translate accelerated-test results into normal-use life predictions.

This approach is essential when products are designed for long service lives (years or decades), making it impractical to wait for failures at use conditions. ALT compresses that time by running tests at higher stress, then mathematically projecting back to the use environment.

5.2 Learning Objectives

By the end of this chapter, you will be able to:

Explain the purpose of accelerated life testing and the role of the Acceleration Factor.
Apply the Arrhenius model for temperature-accelerated tests.
Apply the Power Law model for non-thermal stress acceleration.
Fit ALT models using the WeibullR.ALT package.
Interpret probability plots and relationship plots from ALT analyses.
Use parallelization to constrain the shape parameter across stress levels.

5.3 The Acceleration Factor

The Acceleration Factor (AF) quantifies how much faster a product fails under elevated stress compared to use conditions:

\[\text{Life at use conditions} = AF \times \text{Life at stress conditions}\]

The impact of AF on expected test life:

library(WeibullR.ALT)

AF_vals <- seq(1, 10, by = 0.1)
life_normal <- 10000  # assumed normal-use life

plot(AF_vals, life_normal / AF_vals, type = "l", col = "blue", lwd = 2,
     xlab = "Acceleration Factor (AF)",
     ylab = "Expected Life at Stress Conditions",
     main = "Effect of Acceleration Factor on Test Life")
abline(h = life_normal / c(2, 5, 10), col = "gray70", lty = 2)

An AF of 5 means the product fails five times faster under test stress — a 10,000-hour life in the field becomes a 2,000-hour test.

5.4 The Arrhenius Model

The Arrhenius Model describes the relationship between temperature and reaction rate, which maps directly to material degradation and failure:

\[AF = \exp\left(\frac{E_a}{k}\left(\frac{1}{T_{\text{use}}} - \frac{1}{T_{\text{stress}}}\right)\right)\]

Where: - $E_a$ = activation energy in eV (electron volts) - $k$ = Boltzmann constant ($8.617 \times 10^{-5}$ eV/K) - $T_{\text{use}}$, $T_{\text{stress}}$ = temperatures in Kelvin

Higher activation energies mean the failure mechanism is more sensitive to temperature — a small temperature increase produces a large change in failure rate.

Choosing Activation Energy

$E_a$ is typically drawn from published literature values for the specific failure mechanism, or estimated from pilot tests at multiple temperatures. Representative ranges:

Failure mechanism	Typical $E_a$ (eV)
Corrosion / electromigration	0.5 – 1.0
Electrical insulation degradation	0.4 – 1.5
Semiconductor oxide breakdown	0.3 – 0.7
Thermal fatigue (solder)	0.2 – 0.5

If $E_a$ is unknown, use a sensitivity analysis across a plausible range before drawing conclusions.

WeibullR.ALT: Arrhenius-Lognormal Example

The WeibullR.ALT package (Silkworth 2022) extends WeibullR with ALT models. Data: Class B insulation tested at three oven temperatures (170, 190, 220°C), from Nelson (1990) Table 4.1.

t41    <- NelsonData("table4.1")
set170 <- alt.data(t41$C170f, s = t41$C170s, stress = 170)
set190 <- alt.data(t41$C190f, s = t41$C190s, stress = 190)
set220 <- alt.data(t41$C220f, s = t41$C220s, stress = 220)

arrobj <- alt.make(list(set170, set190, set220),
                   dist = "lognormal", alt.model = "arrhenius",
                   method.fit = "mle", view_dist_fits = FALSE)

The fitted curves shift left as temperature increases — higher temperature leads to shorter life.

Calculating the AF at 220°C vs 170°C with $E_a = 1.0$ eV:

E_a      <- 1.0
k        <- 8.617e-5
T_use    <- 170 + 273  # 443 K
T_stress <- 220 + 273  # 493 K

AF <- exp(E_a / k * (1 / T_use - 1 / T_stress))
cat("Acceleration Factor at 220°C vs 170°C:", round(AF, 1), "\n")

Acceleration Factor at 220°C vs 170°C: 14.3

As printed above, the insulation degrades approximately 14 times faster at 220°C than at 170°C — a substantial acceleration that compresses a multi-year test into a much shorter duration.

Try It

A component is tested at 85°C to accelerate failures. The use temperature is 25°C. The activation energy is 0.65 eV. Compute the AF.

E_a      <- 0.65
k        <- 8.617e-5
T_use    <- 25 + 273.15
T_stress <- 85 + 273.15
# AF = exp((E_a / k) * (1/T_use - 1/T_stress))

Solution

E_a      <- 0.65
k        <- 8.617e-5
T_use    <- 25 + 273.15
T_stress <- 85 + 273.15
AF <- exp((E_a / k) * (1 / T_use - 1 / T_stress))
cat("Acceleration Factor:", round(AF, 1), "\n")

Acceleration Factor: 69.3

5.5 The Power Law Model

The Power Law Model describes how a product’s life is affected by non-thermal stresses (voltage, mechanical load, pressure):

\[AF = \left(\frac{S_{\text{stress}}}{S_{\text{use}}}\right)^{n}\]

Where $n$ is the stress exponent — the sensitivity of the failure mechanism to the applied stress. Typical values range from 1–3 for low-sensitivity mechanisms to 4–8 for voltage-driven dielectric breakdown. $n$ is estimated from the ALT data by fitting the model at multiple stress levels.

WeibullR.ALT: Power Law Example

Data: electrical insulating fluids tested at 7 voltage levels (26–38 kV).

data_nelson <- NelsonData("table3.1")

data26 <- alt.data(data_nelson$kV26, stress = 26)
data28 <- alt.data(data_nelson$kV28, stress = 28)
data30 <- alt.data(data_nelson$kV30, stress = 30)
data32 <- alt.data(data_nelson$kV32, stress = 32)
data34 <- alt.data(data_nelson$kV34, stress = 34)
data36 <- alt.data(data_nelson$kV36, stress = 36)
data38 <- alt.data(data_nelson$kV38, stress = 38)

pwrobj <- alt.make(list(data26, data28, data30, data32, data34, data36, data38),
                   dist = "weibull", alt.model = "power")

Curves shift left as voltage increases — higher voltage stress leads to shorter life.

5.6 Multiple Stress Factors

The Eyring Model extends the Arrhenius approach to account for multiple stress factors (temperature, voltage, humidity):

\[\text{Life}(S) = A \cdot \exp\left(\frac{B}{S} + C \cdot S\right)\]

The formula above assumes a single primary stress $S$. For truly multiple independent stresses (e.g., temperature and humidity and voltage), additional terms are added to the exponent. The WeibullR.ALT package supports the Eyring model via alt.make(..., alt.model = "eyring"). See the package documentation for examples.

5.7 Parallelization

Parallelization constrains the shape parameter (slope) to be equal across all stress levels, simplifying the analysis:

prlobj <- alt.parallel(arrobj)

expobj <- alt.parallel(pwrobj)

With parallel slopes, the relationship between stress and life can be characterized by a single line (the relationship plot).

5.8 Relationship Plots

The relationship plot shows how life changes as stress increases (log-linear, stress on x-axis, log-life on y-axis). The slope characterizes the sensitivity of life to stress:

Negative slope: higher stress → shorter life (typical).
Flat slope: stress has little effect.
Positive slope: higher stress → longer life (rare).

lnrobj <- alt.fit(prlobj)
plot(lnrobj, suppress.dev.new = TRUE)

Each line corresponds to a different probability of failure (e.g., P10 = 10% failure probability, P63.2 = 63.2% failure probability = characteristic life).

lnrobj2 <- alt.fit(expobj)
plot(lnrobj2, suppress.dev.new = TRUE)

Both relationship plots confirm a negative slope — higher stress leads to shorter life in both datasets.

Review

What does a negative slope in a relationship plot indicate?

Answer

Higher stress leads to shorter life. This is the typical finding in ALT — stress accelerates the failure mechanism.

5.9 Important Limitations of ALT

When ALT Predictions May Be Invalid

ALT is a powerful tool, but it rests on a critical assumption: the same failure mechanism operates at elevated stress as at use conditions. If high stress activates a different failure mode — one that would not appear during normal operation — the accelerated results will not transfer to real-world life predictions.

Common pitfalls:

New failure modes at high stress — e.g., thermal cracking from extreme temperature that would never occur at use temperature.
Mechanism change — e.g., ductile failure at low stress transitions to brittle fracture at high stress; the Weibull shape and the AF model no longer apply.
Unknown activation energy or stress exponent — small errors in $E_a$ or $n$ produce large errors in predicted AF; always perform a sensitivity analysis.

When in doubt: run tests at the lowest stress level possible that still produces observable failures in a reasonable time, and verify that failures look the same as field failures.

5.10 Summary

Key takeaways:

Acceleration Factor: $\text{Life}_{\text{use}} = AF \times \text{Life}_{\text{stress}}$.
Arrhenius: temperature-based acceleration; $AF = \exp(E_a/k \cdot (1/T_{\text{use}} - 1/T_{\text{stress}}))$.
Power Law: non-thermal stress acceleration; $AF = (S_{\text{stress}}/S_{\text{use}})^n$.
Parallelization: constrains slope to be equal across stress levels; required before fitting the relationship.
Relationship plots: visualize life vs stress; negative slope = higher stress → shorter life.
Use alt.data() → alt.make() → alt.parallel() → alt.fit() for the full workflow.

Meeker, William Q., and Luis A. Escobar. 1998. Statistical Methods for Reliability Data. Wiley-Interscience.

Nelson, Wayne. 1990. Accelerated Testing. Wiley-Interscience.

Silkworth, David. 2022. WeibullR.ALT: Accelerated Life Testing Using ’WeibullR’. https://doi.org/10.32614/CRAN.package.WeibullR.ALT.

# Accelerated Life Testing {#sec-alt} ## Introduction **Accelerated Life Testing (ALT)** [@Meeker1998; @Nelson1990] subjects products to elevated stress levels — temperature, voltage, mechanical load, humidity — to induce failures more quickly than would occur under normal operating conditions. The acceleration relationship is then used to translate accelerated-test results into normal-use life predictions. This approach is essential when products are designed for long service lives (years or decades), making it impractical to wait for failures at use conditions. ALT compresses that time by running tests at higher stress, then mathematically projecting back to the use environment. ## Learning Objectives By the end of this chapter, you will be able to: - Explain the purpose of accelerated life testing and the role of the Acceleration Factor. - Apply the Arrhenius model for temperature-accelerated tests. - Apply the Power Law model for non-thermal stress acceleration. - Fit ALT models using the `WeibullR.ALT` package. - Interpret probability plots and relationship plots from ALT analyses. - Use parallelization to constrain the shape parameter across stress levels. ## The Acceleration Factor The **Acceleration Factor (AF)** quantifies how much faster a product fails under elevated stress compared to use conditions: $$\text{Life at use conditions} = AF \times \text{Life at stress conditions}$$ The impact of AF on expected test life: ```{r} #| message: false library(WeibullR.ALT) ``` ```{r} AF_vals <- seq(1, 10, by = 0.1) life_normal <- 10000 # assumed normal-use life plot(AF_vals, life_normal / AF_vals, type = "l", col = "blue", lwd = 2, xlab = "Acceleration Factor (AF)", ylab = "Expected Life at Stress Conditions", main = "Effect of Acceleration Factor on Test Life") abline(h = life_normal / c(2, 5, 10), col = "gray70", lty = 2) ``` An AF of 5 means the product fails five times faster under test stress — a 10,000-hour life in the field becomes a 2,000-hour test. ## The Arrhenius Model The **Arrhenius Model** describes the relationship between temperature and reaction rate, which maps directly to material degradation and failure: $$AF = \exp\left(\frac{E_a}{k}\left(\frac{1}{T_{\text{use}}} - \frac{1}{T_{\text{stress}}}\right)\right)$$ Where: - $E_a$ = activation energy in eV (electron volts) - $k$ = Boltzmann constant ($8.617 \times 10^{-5}$ eV/K) - $T_{\text{use}}$, $T_{\text{stress}}$ = temperatures in Kelvin Higher activation energies mean the failure mechanism is more sensitive to temperature — a small temperature increase produces a large change in failure rate. ::: {.callout-note} ## Choosing Activation Energy $E_a$ is typically drawn from published literature values for the specific failure mechanism, or estimated from pilot tests at multiple temperatures. Representative ranges: | Failure mechanism | Typical $E_a$ (eV) | |---|:---:| | Corrosion / electromigration | 0.5 – 1.0 | | Electrical insulation degradation | 0.4 – 1.5 | | Semiconductor oxide breakdown | 0.3 – 0.7 | | Thermal fatigue (solder) | 0.2 – 0.5 | If $E_a$ is unknown, use a sensitivity analysis across a plausible range before drawing conclusions. ::: ### WeibullR.ALT: Arrhenius-Lognormal Example The `WeibullR.ALT` package [@WeibullRalt] extends WeibullR with ALT models. Data: Class B insulation tested at three oven temperatures (170, 190, 220°C), from Nelson (1990) Table 4.1. ```{r} t41 <- NelsonData("table4.1") set170 <- alt.data(t41$C170f, s = t41$C170s, stress = 170) set190 <- alt.data(t41$C190f, s = t41$C190s, stress = 190) set220 <- alt.data(t41$C220f, s = t41$C220s, stress = 220) arrobj <- alt.make(list(set170, set190, set220), dist = "lognormal", alt.model = "arrhenius", method.fit = "mle", view_dist_fits = FALSE) ``` The fitted curves shift left as temperature increases — higher temperature leads to shorter life. Calculating the AF at 220°C vs 170°C with $E_a = 1.0$ eV: ```{r} E_a <- 1.0 k <- 8.617e-5 T_use <- 170 + 273 # 443 K T_stress <- 220 + 273 # 493 K AF <- exp(E_a / k * (1 / T_use - 1 / T_stress)) cat("Acceleration Factor at 220°C vs 170°C:", round(AF, 1), "\n") ``` As printed above, the insulation degrades approximately 14 times faster at 220°C than at 170°C — a substantial acceleration that compresses a multi-year test into a much shorter duration. ::: {.callout-note} ## Try It A component is tested at 85°C to accelerate failures. The use temperature is 25°C. The activation energy is 0.65 eV. Compute the AF. ```{r} E_a <- 0.65 k <- 8.617e-5 T_use <- 25 + 273.15 T_stress <- 85 + 273.15 # AF = exp((E_a / k) * (1/T_use - 1/T_stress)) ``` <details><summary>Solution</summary> ```{r} E_a <- 0.65 k <- 8.617e-5 T_use <- 25 + 273.15 T_stress <- 85 + 273.15 AF <- exp((E_a / k) * (1 / T_use - 1 / T_stress)) cat("Acceleration Factor:", round(AF, 1), "\n") ``` </details> ::: ## The Power Law Model The **Power Law Model** describes how a product's life is affected by non-thermal stresses (voltage, mechanical load, pressure): $$AF = \left(\frac{S_{\text{stress}}}{S_{\text{use}}}\right)^{n}$$ Where $n$ is the **stress exponent** — the sensitivity of the failure mechanism to the applied stress. Typical values range from 1–3 for low-sensitivity mechanisms to 4–8 for voltage-driven dielectric breakdown. $n$ is estimated from the ALT data by fitting the model at multiple stress levels. ### WeibullR.ALT: Power Law Example Data: electrical insulating fluids tested at 7 voltage levels (26–38 kV). ```{r} data_nelson <- NelsonData("table3.1") data26 <- alt.data(data_nelson$kV26, stress = 26) data28 <- alt.data(data_nelson$kV28, stress = 28) data30 <- alt.data(data_nelson$kV30, stress = 30) data32 <- alt.data(data_nelson$kV32, stress = 32) data34 <- alt.data(data_nelson$kV34, stress = 34) data36 <- alt.data(data_nelson$kV36, stress = 36) data38 <- alt.data(data_nelson$kV38, stress = 38) pwrobj <- alt.make(list(data26, data28, data30, data32, data34, data36, data38), dist = "weibull", alt.model = "power") ``` Curves shift left as voltage increases — higher voltage stress leads to shorter life. ## Multiple Stress Factors The **Eyring Model** extends the Arrhenius approach to account for multiple stress factors (temperature, voltage, humidity): $$\text{Life}(S) = A \cdot \exp\left(\frac{B}{S} + C \cdot S\right)$$ The formula above assumes a single primary stress $S$. For truly multiple independent stresses (e.g., temperature *and* humidity *and* voltage), additional terms are added to the exponent. The `WeibullR.ALT` package supports the Eyring model via `alt.make(..., alt.model = "eyring")`. See the package documentation for examples. ## Parallelization **Parallelization** constrains the shape parameter (slope) to be equal across all stress levels, simplifying the analysis: ```{r} prlobj <- alt.parallel(arrobj) expobj <- alt.parallel(pwrobj) ``` With parallel slopes, the relationship between stress and life can be characterized by a single line (the relationship plot). ## Relationship Plots The **relationship plot** shows how life changes as stress increases (log-linear, stress on x-axis, log-life on y-axis). The slope characterizes the sensitivity of life to stress: - **Negative slope**: higher stress → shorter life (typical). - **Flat slope**: stress has little effect. - **Positive slope**: higher stress → longer life (rare). ```{r} lnrobj <- alt.fit(prlobj) plot(lnrobj, suppress.dev.new = TRUE) ``` Each line corresponds to a different probability of failure (e.g., P10 = 10% failure probability, P63.2 = 63.2% failure probability = characteristic life). ```{r} lnrobj2 <- alt.fit(expobj) plot(lnrobj2, suppress.dev.new = TRUE) ``` Both relationship plots confirm a negative slope — higher stress leads to shorter life in both datasets. ::: {.callout-tip} ## Review What does a negative slope in a relationship plot indicate? <details><summary>Answer</summary> Higher stress leads to **shorter life**. This is the typical finding in ALT — stress accelerates the failure mechanism. </details> ::: ## Important Limitations of ALT ::: {.callout-important} ## When ALT Predictions May Be Invalid ALT is a powerful tool, but it rests on a critical assumption: **the same failure mechanism operates at elevated stress as at use conditions**. If high stress activates a different failure mode — one that would not appear during normal operation — the accelerated results will not transfer to real-world life predictions. Common pitfalls: - **New failure modes at high stress** — e.g., thermal cracking from extreme temperature that would never occur at use temperature. - **Mechanism change** — e.g., ductile failure at low stress transitions to brittle fracture at high stress; the Weibull shape and the AF model no longer apply. - **Unknown activation energy or stress exponent** — small errors in $E_a$ or $n$ produce large errors in predicted AF; always perform a sensitivity analysis. When in doubt: run tests at the *lowest* stress level possible that still produces observable failures in a reasonable time, and verify that failures look the same as field failures. ::: ## Summary **Key takeaways:** - **Acceleration Factor**: $\text{Life}_{\text{use}} = AF \times \text{Life}_{\text{stress}}$. - **Arrhenius**: temperature-based acceleration; $AF = \exp(E_a/k \cdot (1/T_{\text{use}} - 1/T_{\text{stress}}))$. - **Power Law**: non-thermal stress acceleration; $AF = (S_{\text{stress}}/S_{\text{use}})^n$. - **Parallelization**: constrains slope to be equal across stress levels; required before fitting the relationship. - **Relationship plots**: visualize life vs stress; negative slope = higher stress → shorter life. - Use `alt.data()` → `alt.make()` → `alt.parallel()` → `alt.fit()` for the full workflow.