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Reliability Physics and Engineering
The timetofailure models which were developed in the previous chapters assume that the stress remains constant with time until the material fails. Even when we discussed fatigue (a failure mechanism caused by a cyclical stress), it was assumed that the stress range Δσ remained constant with time. However, seldom is the applied stress constant with time, as illustrated in Fig. 14.1. In integrated circuits, the currents and fields are continually changing during operation and generally depend on the frequency of operation. In mechanical devices, the mechanical stress usually varies with time (the mechanical stress in a metal light pole changes with wind direction and with wind speed while the mechanical stress in the shaft of a rotor changes with the number of rpm). Therefore, a question naturally arises: how does one convert dynamical stresses (timedependent stresses) ξ(t) into an effective static form ξeffective so that all of the previously developed timetofailure models can be used? This chapter presents a methodology for that conversion.
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1.
The electric field
E in a capacitor dielectric is expected to operate at 4 MV/cm during a period of 16 ns. However, during this period, a sharp rise/pulse in the electric field (rising from 4 to 8 MV/cm) occurs between 4 and 7 ns. Using the fullwidthathalfmaximum approach for the pulse, calculate the effective constant electric field for the 16 ns period shown. Assume an exponential field acceleration parameter of
γ = 4.0 cm/MV.
×
Answer:
E
_{effective} = 7.48 MV/cm
2.
A mechanical component experiences a timedependent tensilestress waveform given by:
$$ \sigma (t)=\frac{1.863\times {10}^{5}\mathrm{MPa}}{{\left(\mathrm{h}\right)}^9}{(t)}^9\exp \left[{\left(\frac{t}{8}\right)}^{10}\right]. $$
The shape of the waveform is shown below.
×
Assuming a creep powerlaw exponent of
n = 4:
(a)
Find the effective rectangular pulse over the time interval from 4 h to 10 h.
(b)
Assuming the period is 24 h, what is the effective constant value of the stress over this period?
Answer: (a) 599 MPa (b) 424 MPa
3.
For wind turbine use, the energy contained in the wind is a critically important parameter. The energy contained in the wind is proportional to the square of the wind speed. For Dallas, Texas, the following mean wind speeds were reported by month:
(a)
Find the mean value for the wind speed
S for the entire year.
Month

# Days

Wind speed: S (mph)

(Duty cycle)
_{i}


January

31

11.0

0.085

February

28

11.7

0.077

March

31

12.6

0.085

April

30

12.4

0.082

May

31

11.1

0.085

June

30

10.6

0.082

July

31

9.8

0.085

August

31

8.9

0.085

September

30

9.3

0.082

October

31

9.7

0.085

November

30

10.7

0.082

December

31

10.8

0.085

(b)
Given the energy in the wind goes as the square of the wind speed, find the constant effective wind speed (
S)
_{effective} for turbine use during the entire year.
Answers:
(a)
Mean Speed = 10.7 mph
(b)
(
S)
_{effective} = 10.8 mph
4.
The mission profile is shown below for a mechanical component. Assuming that the mechanical component is a metal that has no yield point and a powerlaw of
n = 4 for creep, find the effective constant stress
σ
_{effective} for the full 10 years (120 Months) of service.
Stress level:
σ (MPa)

Time (months)

(Duty cycle)
_{i}


0

1

0.008

100

2

0.017

200

4

0.033

300

6

0.050

400

18

0.150

500

35

0.292

600

25

0.208

700

15

0.125

800

9

0.075

900

4

0.033

1,000

1

0.008

Total = 120

1.000

Answer:
σ
_{effective} = 612 MPa
5.
Using the mission profile for the metal component in Problem 4, what would be the effective constantstress value
σ
_{effective} for the full 10 years (120 months) of service if the metal component has a powerlaw stress exponent of
n = 4 for creep and has a yield strength of 400 MPa?
Answer:
σ
_{effective} −
σ
_{yield} = 283 MPa or
σ
_{effective} = 683 MPa
6.
Using the mission profile for the metal component in Problem 4, what would be the effective constantstress value
σ
_{ffective} for the full 10 years (120 months) of service if a powerlaw exponent of
n = 6 for creep is assumed and no defined yield strength?
Answer:
σ
_{effective} = 650 MPa
7.
Using the mission profile for the metal component in Problem 4, what would be the effective constantstress value
σ
_{effective} for the full 10 years (120 months) of service if a powerlaw exponent of
n = 6 for creep is assumed and a yield strength of 400 MPa?
Answer:
σ
_{effective} −
σ
_{yield} = 331 MPa or
σ
_{effective} = 731 MPa
8.
EM is a concern for a certain conductor. The current densities in the conductor are shown below. What is the average current density?
Current density:
J (A/cm
^{2})

(Duty cycle)
_{i}


5.00E + 05

0.4

7.00E + 05

0.3

9.00E + 05

0.2

1.20E + 06

0.1

Answer: (
J)
_{average} = 7.1 × 10
^{5} A/cm
^{2}
9.
The mission profile for a component, with fatigue concerns, is shown below. Assuming a powerlaw exponent of
n = 4 and no elastic range, find the effective constant value for the stress range (Δ
σ)
_{effective}.
Stress range: Δ
σ (MPa)

(Duty cycle)
_{i}


300

0.10

400

0.25

500

0.36

600

0.23

700

0.05

800

0.01

Answer: (Δ
σ)
_{effective} = 524 MPa
10.
A silicabased capacitor dielectric of thickness 45Å will see the following voltages during operation. What is the effective constant voltage
V
_{eff} for TDDB? Assume an exponential field acceleration parameter of
γ = 4.0 cm/ MV.
Voltage (V)

(Duty cycle)
_{i}


2.5

3.00E − 01

2.8

1.50E − 01

3.1

1.00E − 01

3.4

4.00E − 02

3.7

2.00E − 01

4.0

2.00E − 01

4.3

1.00E − 02

×
Answer:
E
_{effective} = 7.48 MV/cm
A mechanical component experiences a timedependent tensilestress waveform given by:
$$ \sigma (t)=\frac{1.863\times {10}^{5}\mathrm{MPa}}{{\left(\mathrm{h}\right)}^9}{(t)}^9\exp \left[{\left(\frac{t}{8}\right)}^{10}\right]. $$
×
Assuming a creep powerlaw exponent of
n = 4:
(a)
Find the effective rectangular pulse over the time interval from 4 h to 10 h.
(b)
Assuming the period is 24 h, what is the effective constant value of the stress over this period?
(a)
Find the mean value for the wind speed
S for the entire year.
Month

# Days

Wind speed: S (mph)

(Duty cycle)
_{i}


January

31

11.0

0.085

February

28

11.7

0.077

March

31

12.6

0.085

April

30

12.4

0.082

May

31

11.1

0.085

June

30

10.6

0.082

July

31

9.8

0.085

August

31

8.9

0.085

September

30

9.3

0.082

October

31

9.7

0.085

November

30

10.7

0.082

December

31

10.8

0.085

(b)
Given the energy in the wind goes as the square of the wind speed, find the constant effective wind speed (
S)
_{effective} for turbine use during the entire year.
Month

# Days

Wind speed: S (mph)

(Duty cycle)
_{i}


January

31

11.0

0.085

February

28

11.7

0.077

March

31

12.6

0.085

April

30

12.4

0.082

May

31

11.1

0.085

June

30

10.6

0.082

July

31

9.8

0.085

August

31

8.9

0.085

September

30

9.3

0.082

October

31

9.7

0.085

November

30

10.7

0.082

December

31

10.8

0.085

(a)
Mean Speed = 10.7 mph
(b)
(
S)
_{effective} = 10.8 mph
Stress level:
σ (MPa)

Time (months)

(Duty cycle)
_{i}


0

1

0.008

100

2

0.017

200

4

0.033

300

6

0.050

400

18

0.150

500

35

0.292

600

25

0.208

700

15

0.125

800

9

0.075

900

4

0.033

1,000

1

0.008

Total = 120

1.000

Current density:
J (A/cm
^{2})

(Duty cycle)
_{i}


5.00E + 05

0.4

7.00E + 05

0.3

9.00E + 05

0.2

1.20E + 06

0.1

Stress range: Δ
σ (MPa)

(Duty cycle)
_{i}


300

0.10

400

0.25

500

0.36

600

0.23

700

0.05

800

0.01

11.
The thermal profile for a device is shown below. Assuming an activation energy of 0.7 eV, what is the effective constanttemperature
T
_{effective}?
Temp (°C)

Temp (K)

Time (months)

Duty cycle (dyc)


180

453

1

0.00833

150

423

7

0.05833

125

398

12

0.10000

95

368

70

0.58333

75

348

24

0.20000

25

298

6

0.05

 Titel
 Conversion of Dynamical Stresses into Effective Static Values
 DOI
 https://doi.org/10.1007/9783319936833_14
 Autor:

J. W. McPherson
 Sequenznummer
 14
 Kapitelnummer
 Chapter 14