4.2.1 Acceleration Factor Modification of Miner’s Damage Rule

It can be difficult to always know the cycles to failure at any ith stress level in Miner’s application. However, since we are on the same work path, we can often establish a cyclic acceleration factor (see Special Topic B for examples) between stress levels. If AF_damage and N₁ are known, then Miner’s rule can be simplified from Equation (4.9) for the ith stress using

(4.10)

Then Miner’s rule can be modified as [3, 4]:

(4.11)

The acceleration factor AF_damage is given in Section 4.3.4 and depends on the type of vibration (sine or random). See also Section 4.5 for use of this equation in fatigue damage spectrum (FDS) analysis. It can also be used in thermal cycling (see Section 4.3.3).

4.3.1 Example 4.2: Creep Cumulative Damage and Acceleration Factors

Creep parameters include the strain (ε) length change ΔL (ΔL/L) due to an applied stress (σ) at temperature (T). In the elastic region, stress causes a strain that is recoverable (i.e., reversible) so that where Y is Young’s modulus. When stress increases such that the work is irreversible, the permanent plastic strain ε_p damage is termed plastic deformation. The most popular empirical creep rate equation is [3, 5]

(4.12)

where B₀ is a material strength constant; p is the time exponent where 0 < p < 1 for primary and secondary creep stages; M (where 0 < 1/M < 1) is strain hardening exponent, dependent on the material type; and E_a is the thermal activation energy for the creep process (see Figure 9.5 for the three stages of creep). We note a linear time dependence is observed in the secondary creep phase where p = 1. Some tabulated values are given in Table 4.1 for M and B₀ for the secondary creep rate.

The thermodynamic work causing damage from the creep process of the metal is found from the stress–strain creep area when the stress–strain relation is plotted in Figure 4.1 to demonstrate typical creep data.

The conjugate work variables for creep are stress σ for force and ε = ΔL/L strain, the length variable. From Figure 4.1 displaying creep, we can see that it is logical to model the strain over time by a power law; the strain will also be a power law as a function of stress. We know from Equation (4.12) that these are good assumptions (see Figure 4.1), so that our general expression is

(4.13)

Furthermore, from Figure 4.1 it is logical that 0 < M < 1 for primary–secondary creep region (Figure 4.2).

The thermodynamic work causing damage from the creep process depicted in Figure 4.2 is found from the stress–strain creep area when the stress–strain relation is plotted. Assessing the damage is more accurately found if the data were available. Here, we can use the empirical creep rate expression to find this area by integrating the expression above to determine the work in terms of the applied stress, which is more easily known. The creep path is dL (ε = ΔL/L) for the integral limits, so summing the thermodynamic work we have

(4.14)

Making comparisons to the original empirical creep equation (Equation (4.12)), we have A(T) is defined within a constant , while and Y = p. We have assumed that stress is not time dependent. Note that we prefer to use the term dε/dt for a constant stress to integrate over time to obtain damage in terms of the applied stress.

In order to assess the damage, we need to have some knowledge of the critical damage at a particular stress and temperature. Let’s assume this occurs at time τ₁ at stress level σ₁ and temperature T₁. Then the thermodynamic damage ratio at any other stress σ₂ and temperature T₂ at time t₂ along the same work path is

(4.15)

where and . Here, τ_i is time to failure (a constant) at stress i and t_i is work time at stress level i. Again, this is valid for stresses 1, 2 along the same work path ΔL. If damage is 1, failure occurs (at t₂ = τ₂) and we can then write this as

(4.16)

This is the creep acceleration factor (see Special Topic B on Acceleration Factor usage). Recall that (i.e., work to failure stress 2 and 1; see Equation (4.6)), so their ratio will be 1. Here M/p = K and Q = E_a/p. Often it is helpful to write the linear form for time to failure for a particular stress. This is deduced as follows from the above equation

(4.17)

Writing the time to failure t_failure for τ₂ we have

(4.18)

where the constant C can always be written in terms of another stress if we know that ; however, it also equates to the factored-out constant in Equation (4.15) C = ln(1/A).

Lastly, it should be noted that if a number of different i stresses are applied, the general creep damage ratio can be written by accumulating the thermodynamic damage along the same work path (Equation (4.15)) as

(4.19)

In Section 4.5 and Special Topics B.7 we illustrate how this equation can be used for environmental profiling in order to find a cumulative accelerated stress test (CAST) goal.

4.3.2 Example 4.3: Wear Cumulative Damage and Acceleration Factors

Wear nicely exemplifies the thermodynamic approach. There are many different types of wear, including abrasive, adhesive, fretting, and fatigue wear. The most common wear model used for adhesive and sometimes abrasive wear of the softer material between two sliding surfaces is the Archard’s wear equation [6, 7]:

(4.20)

where D is removed depth of the softer material; P_W is normal load (lb); l is the sliding distance (feet); H is hardness of the softer material (psi); A is contact area; and k is Archard’s wear coefficient (dimensionless). In the adhesive wear of metals [6–8], wear coefficient k varies between 10⁻⁷ and 10⁻² depending on the operating conditions and material properties. It should be recognized that a wear coefficient k is constant typically only within a certain adhesive wear-rate range.

We define this as our system as shown in Figure 4.3, consisting of the weight, contact area, hardness, and k value and which is traveling at some velocity. Next we need to provide the environment which is responsible for doing work on our system which causes the movement so that wear can occur. We introduce an external force P_E as shown in Figure 4.3.

P_F creates the sliding and induces the thermodynamic wear work. Some of the external work goes into creating the wear velocity while some causes wear. Work is simply the integral of the force times distance or and dx is the sliding distance. It is logical that P_F is proportional to P_W, but since we are causing wear we write it as . The distance dx can be written as , where C₂ is a constant related to the surface wear friction

(4.21)

Note that the integral is over x₁ to x₂, the work path in the time t₁ to t₂, which is linear with wear amount. Since we have substituted vt for the sliding distance, we need to keep this in mind. Here we have introduced the knowledge of Archard’s wear model by identifying that our constant C₂ is k/AH times some constant that is related to the work type. This constant is needed as without it we would just have Archard’s equation which has units of distance not work.

At some time t = τ₁ we may consider that too much damage has occurred in a stress environment we call 1 as compared to stress environment 2. Then we can assess the damage ratio between environments 1 and 2, where 1 has caused failure and in 2 failure has not yet occurred:

(4.22)

where τ₁ is the time for critical wear failure in stress environment 1. Note that for the two environments (1 and 2) the materials are the same so H and k cancel out. We must also stay on the same work type to make comparisons between environments; therefore k_P are the same.

When failure occurs (the amount of wear is the same for both environments), the thermodynamic work in environment 2 equals that of environment 1, the damage ratio is 1, and . This occurs at time τ₂ in environment 2, so we can then write the wear acceleration factor as

(4.23)

(See Special Topics B for how to apply acceleration factors in testing.) The subscript DF indicates that we have created the same wear amount causing failure in both environments. One thing we note is that Archard’s type of wear is not easily accelerated as it is linear with weight and velocity.

It is helpful to write the linear form for time to failure for a particular stress. This is deduced to within a constant from the above equation

(4.24)

Then the time to failure for wear can be written t_failure = τ:

(4.25)

where C is a constant that can be written in terms of another stress level if known. However, it also represents the ratio of the factored-out constant .

If a number of different i stresses are applied along the same work path (Equation (4.22)). The general wear damage ratio can be written by accumulating the thermodynamic damage at each stress level as

(4.26)

In Section 4.3 and Special Topic B.7 we illustrate how this equation can be used for environmental profiling in order to find a CAST goal.

If the external force causes oscillator motion so that we are doing cyclic work for n cycles, we can replace t by n and τ by N to obtain Miner’s Rule for Archard’s type wear.

4.3.3 Example 4.4: Thermal Cycle Fatigue and Acceleration Factors

In thermal cycling, a temperature change ΔT in the environment from one extreme to another causes expansion and/or contraction (i.e., strain) in a material system. The plastic strain (ε) caused by the thermal cyclic stress (σ) in the material can be written similar to Equation (4.12) as:

(4.27)

where we have substituted σ^M ~ ΔT ^b for the non-linear stress and n for thermal cycles instead of time.

The thermodynamic work causing damage from thermal cycle stress is found from the stress–strain creep area if the stress–strain relation could be plotted. Then, similar to creep work above, the cyclic work is still over the path ΔL to Equation (4.14). However, the work path is not much of an issue in say solder joint expansion contraction or other joints that may have different expansion–contraction rates. The main concern is keeping the system, such as the solder joint, consistent during analysis. We have

(4.28)

where . We need to have some knowledge of the critical damage at a particular cyclic stress and along this work path. Let’s assume this occurs at N₁ at stress level ΔT₁. Then, similar to Equation (4.15), the damage ratio at another stress ΔT₂ for n₂ cycles is

(4.29)

If damage is 1 then , failure occurs and we write the acceleration factor

(4.30)

where and . The non-Arrhenius ratio is called the “Coffin-Manson” acceleration factors [9–11], or Equation (4.30) is the Modified Coffin-Manson acceleration factor. When the activation energy E_a is small then the Arrhenius effect can be neglected. For example, in solder joint testing, K = 1.9 for lead-free solder and about 2.5 for lead solder. The activation energy is about 0.123 eV that is typically used. For example, if use condition is stress level 1 cycle between 20 and 60 (ΔT = 40°C, T_max = 60°C), while test stress condition is stress level 2 cycled between −20 and 100°C (ΔT = 120°C, T_max = 100°C), then Arrhenius AF = 1.58 while the Coffin-Manson AF = 9 with an overall AF of 14.2. In the case where we have 1 cycle per day in use condition, we see that 10 years of use condition is about 260 test cycles (see Special Topics B for more applications).

Equation (4.30) is also similar to the Norris–Lanzberg [12] thermal cycle model which also includes a thermal cycle frequency effect (see Equation (B18) for details and uses of their model).

It is helpful to write the linear form for cycles to failure for a particular stress. This is deduced as

(4.31)

The thermal cycles to failure is therefore

(4.32)

where C can be written in terms of the other stress factor if known, . However, it can also be found as the factored-out ratio constant C = ln(1/B).

Lastly, it should be noted that if a number of different stresses are applied the general damage ratio can be written by accumulating the thermodynamic damage along the same work path (Equation (4.29)) as in Miner’s rule:

(4.33)

In Section 4.4 and Special Topics B.7 we illustrate how this equation can be used for environmental profiling in order to find a CAST goal.

4.3.4 Example 4.5: Mechanical Cycle Vibration Fatigue and Acceleration Factors

In a similar manner to the above argument for thermal cycle, we can find the equivalent mechanical vibration cyclic fatigue damage. In a vibration environment, a vibration level depends on the type of exposure. In testing two types of environments are typically used: sinusoidal and random vibration profiles. In sinusoidal vibration the stress level is denoted G_s, where G is a unitless quantity equal to the sinusoidal acceleration A divided by the gravitational constant g. In random vibration, a similar quantity is used termed G_rms (defined below). Consider first the plastic strain (ε) caused by a sinusoidal vibration level G stress (σ) in the material. The strain can be written similarly to Equation (4.27):

(4.34)

The cyclic work is then found, similar to Equation (4.28), as

(4.35)

where Y = j + 1. Similar to the above arguments, to assess the damage we need to have some knowledge of the critical damage at a vibration stress. Let’s assume this occurs at N₁ at stress level G₁. Then, as in Equation (4.29), the thermodynamic damage ratio at any other stress G₂ level at n₂ cycle is

(4.36)

If damage is 1, n₂ = N₂, and failure occurs. Following the arguments of the other examples, we find the acceleration factor is

(4.37)

where b = Y/P. Since the number of cycles is related to cycle frequency f and the time τ according to

(4.38)

then if f is constant AF_damage is a commonly used relationship for cyclic compression where

(4.39)

This is commonly used for the acceleration factor in sinusoidal testing. For random vibration above, we substitute for G the random vibration G_rms level [13] (see Special Topics B for examples).

It is helpful to write the linear form for cycles to failure for a particular stress. This is deduced to within a constant from the above equation

(4.40)

where is treated as a constant. This is essentially the relation that holds for what is called the S–N curves (see Figure 1.3). Note that if we write the cyclic equation with G ∝ S where S is the stress, we have

(4.41)

where , C is the proportionality constant when going from G to S, and K is a constant similar to C. Note that some authors write this as proportional to the strain instead of the stress. The relationship is generally used to analyze S–N data. b and C are often referred to as Basquin’s equation, commonly written as

(4.42)

Note that since stress and strain are conjugate variables, one can write this alternately in terms of the strain. However, it is usually written this way as the experimental number of cycles to failure N for a given stress S level constitutes S–N curve data in fatigue testing of materials. Such data are widely available in the literature. The slope of the S–N curve (see Figure 1.3) provides an estimate of the exponent b above. S–N data are commonly determined using sinusoidal stress. Often we do not know b. Typical values of b are 4 < b < 8. Some guidance is provided in MIL-STD-810F. For example, it recommends b = 8 for broadband random (discussed below) and b = 6 where for profiles. A conservative test is obtained by setting b = 4. Testing can of course can determine b for a particular failure mode.

Most device vibration testing is typically either sinusoidal or random. The goal is to try and accelerate the type of vibration occurring under use conditions. For automotive, for example, this is random vibration. For a piece of equipment undergoing cyclic motion, this is more likely sinusoidal. The relation for random data is b/2 when using the power spectral density (PSD) (G_rms²/Hz) level instead of G_rms. This is evident from the fact that the G_rms level is found as the square root of the area under the PSD spectrum:

(4.43)

In the simple case of random vibration, white noise for example, we have

(4.44)

Here the time compression expression above for random vibration is over the same bandwidth , then inserting Equation (4.44) into Equation (4.39) we have

(4.45)

This is a common form used for the random vibration acceleration factor [13] (see Special Topics B for examples). The general form for the cycles to failure for a particular stress is then

(4.46)

Finally, it should be noted that if we have applied a number of different stresses, the general damage ratio can be written by accumulating the thermodynamic damage along the same work path (Equation (4.36)) as

(4.47)

This equation turns out to be extremely useful. In Section 4.4 and Special Topics B.7 we illustrate how this equation can be used for environmental profiling in order to find a CAST goal.

4.3.5 Example 4.6: Cycles to Failure under a Resonance Condition: Q Effect

When a product has a built-in resonance that is causing a higher stress level, the fatigue life will be shortened due to magnification of the resonance. In order to model the resonance condition for fatigue life, we first need to explain some common resonance terms to aid the unfamiliar reader. Resonance is commonly quantified using an amplitude magnification factor denoted Q which is typically measured in a number of ways. Figure 4.4 illustrates a resonance. The most common way to measure a resonance is through the transmissibility. The transmissibility is the output divided by the input peak G or G_rms level. This is illustrated for sinusoidal vibration in Figure 4.4, giving

(4.48)

An alternate way to measure Q when the resonance curve is available but the input level is not known is by assessing the resonance value f₀ divided by the resonance width Δf at the maximum amplitude, divided by the square root of 2 as illustrated in Figure 4.4, and given by

(4.49)

Since we have already quantified the cycles to failure in terms of the G or G_rms level, then under a resonance condition the output G level is simply multiplied by the Q magnification level at resonance so the time to failure is

(4.50)

Here G_rms(Q) indicates that the G_rms level is a function of Q. We see the cycles to failure generally goes as the input G-level multiplied by Q for sinusoidal vibration to an inverse power, so that larger Q and/or G values yield a smaller number of cycles to failure. Recall that since time is related to cycles through the frequency of vibration as time = N-cycles/frequency, then we can put this in terms of time to failure if needed.

If we have a random vibration input then the G_rms level is not easily assessed. The simplest instructional way to estimate the G_rms level from the input PSD level at resonance is to use the well-known Miles’ equation [14, 15], defined:

(4.51)

Miles’ equation is applicable for a single degree of freedom. It is an approximate formula that assumes a flat power spectral density in the neighborhood of the resonance f₀. As a rule of thumb, it may be used if the power spectral density is flat over at least two octaves centered at the natural frequency. To ensure narrow band resonance over a flat portion of the spectrum, a Q ~ 10 or higher is often recommended.

As an example for using the Miles’ equation, if we have a single degree of vibration system with an 80 Hz resonance having a Q of 12 and a random 0.04 G_rms²/Hz input level then the G_rms resonance level at resonance is estimated as:

(4.52)

Random vibration is not simple to understand. As the term suggests, the input vibration levels are not constant but are random in nature. Therefore, there is a probability of occurrence in amplitude. The Miles’ equation above is written in terms of a 1-sigma response which means then when the random vibration is roughly Gaussian that this Grms level occurs about 68.27% of the time. Since Miles’ equation stipulates that the PSD be approximately flat near the resonance condition, then this is atypical for a Gaussian-like input in the frequency domain.

Using Miles’ equation, the time to failure can therefore be written for a single degree of freedom where the natural frequency response to a random vibration input can be approximated as above for a Q ∼ 10 or higher and a flat PSD input near f₀ and, under these conditions, will occur about 67% of the time (1-sigma) as

(4.53)

It is important to note that Q is the inverse of damping as

(4.54)

where ξ is the damping factor and η is the loss factor. Often materials are characterized using the loss factor [3] and it is helpful to know which materials can be used to reduce resonances. Table 4.2 provides a list of typical well-known loss factors for different materials.

4.5.1 Fatigue Damage Spectrum for Sine Vibration Accelerated Testing

In sine vibration, we create cyclic fatigue. The input G level will cause a stress on the components and can cause cyclic fatigue that we described in Equation (3.23). The G stress level even in sine vibration will vary with frequency, and Equation (3.25) cyclic damage can be written

(4.55)

If we knew the cyclic work area at each test frequency point, perhaps with the use of a strain gage, we could accumulate the amount of work that was done in some period of time where cycles n = f × (test time). Often it is approximated using Miner’s rule, so we can write the damage over a resonance point due to frequency test from f_min < f_n < f_max as:

(4.56)

where d_i is the damage at the ith stress level which varies with frequency. This of course is an alternate form of Miner’s Rule (Equation (4.9)). For example,

(4.57)

where we have used Equation (4.42) (N = CS^−b) as in Equation (4.11). The stress level will go as the vibration amplitude, denoted here as Z(f) (function of frequency f) to within a K factor, S(f) = KZ(f), so we write the fatigue damage spectra as [17–20]

(4.58)

4.5.1.1 Example 4.7: Fatigue Damage Spectrum for Sine Vibration at and across Resonance

At resonance, we only have one stress level i so that n simply equates to (where t is the test time). Next in sine testing the displacement is defined , where ω₀ is the resonance frequency. The acceleration is the second derivative so the maximum acceleration = Zω₀² and the acceleration in terms of G_s is gG, so that . However, the peak to peak displacement is what is needed so this is a factor of 2 difference; furthermore, since we are at resonance we must take into account the amplification factor Q at resonance (see Figure 4.4) so . Then Miner’s rule for fatigue testing at a sinusoidal resonance yields the following FDS spectrum:

(4.59)

If we are testing across the resonance, then the amplitude is given by

(4.60)

The denominator is found from the harmonic oscillator amplitude equation which is found in numerous books and articles. It is not reproduced here as it is easily referenced.

4.5.2 Fatigue Damage Spectrum for Random Vibration Accelerated Testing

In a very similar manner we can use the concepts for random vibration. We will consider looking at the general damage spectrum (GDS) near a resonance (that will be related to FDS shortly) in random vibration. Then we will have, similar to Equation (4.58),

(4.61)

In random vibration FDS theory, the stress is proportional to the relative displacement of the single degree of freedom (SDOF) (i.e., single axis) multiplied by a constant K:

(4.62)

When the test vibration distribution is Gaussian, Mile’s equation is in many cases a good estimate for the G_rms displacement that takes place. That is, there is a certain probability for displacement amplitudes. Using Mile’s equation (Equation (4.51)), this is:

(4.63)

By inserting Equations (4.63) and (4.62) into (4.61), we have for the general damage fatigue spectrum for any ith SDOF narrow-band random vibration Gaussian

(4.64)

Finally, the FDS differs from the GDS theory by what is called the expected values for the amplitude for a narrow-band Gaussian, which only differs by a gamma numeric factor that goes with b as shown below

(4.65)

These data in the field are collected over the ith W_PSD-input narrow-band Gaussian areas of frequency, as these are worst cases. Then the FDS is generated for the environment. In general, the constants are taken as K = C = 1, the exponent b = 4, 8, or 12 (see discussion in Section 4.3.4) and the amplification factor Q = 10, 25, or 50. The inverse of this equation reproduces the test spectrum and is written

(4.66)

Once we have the test profile, W(f) the actual test-level W(f)-Test is specified by changing the t to t_eq for the test to accelerated time. This is consistent with Equation (4.46) when time is put in terms of cycles to failure. Typical values for b or G according to the Mil-Std 810F are around 8.