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Chapter 8: Problem 31

A cylindrical component constructed from an S-590 alloy (Figure 8.30) has a diameter of 12 \(\mathrm{mm}(0.50\) in.). Determine the maximum load that may be applied for it to survive \(500 \mathrm{~h}\) at \(925^{\circ} \mathrm{C}\left(1700^{\circ} \mathrm{F}\right) .\)

Short Answer

Expert verified
To answer this question, follow these steps: 1. Obtain the creep properties of S-590 alloy, including material constant \(A\), activation energy \(Q\), and stress exponent \(n\). 2. Convert the temperature of 925°C to Kelvin (1198 K). 3. Calculate the cross-sectional area of the cylinder (approximately 113.1 mm²). 4. Find or assume an appropriate creep strain (\(\varepsilon\)) for the given material and conditions. 5. Calculate the initial creep strain rate (\(\dot{\varepsilon}_i\)) using the creep life span of 500 hours and the total creep strain. 6. Solve for the applied stress (\(\sigma\)) using the equation \(\dot{\varepsilon}_i = A\sigma^n e^{\frac{-Q}{RT}} \). 7. Calculate the maximum load (\(P_{max}\)) using the applied stress and the cross-sectional area. Provide your own values for the creep properties and allowable total creep strain to obtain the maximum load value.

Step by step solution

01

Creep Properties of S-590 Alloy

To perform the analysis, we first need to look at the creep properties of the S-590 alloy, which can be found in the provided Figure 8.30. The creep rate is given as a function of the applied stress and temperature. Creep rate decreases as temperature and stress decrease, while a specimen's life span increases with a lower stress and temperature. The relationship between creep strain rate, stress, and temperature can be given by the following equation: \( \dot{\varepsilon} = A\sigma^n e^{\frac{-Q}{RT}} \) where \(\dot{\varepsilon}\) is the creep strain rate, \(A\) is a constant, \(\sigma\) is the applied stress, \(n\) is the stress exponent, \(Q\) is the activation energy for creep, \(R\) is the gas constant, and \(T\) is the temperature in Kelvin. We will use this equation to find the maximum load.
02

Convert Temperature to Kelvin

First, we need to convert the given temperature of 925°C to Kelvin: \(T_{K} = 925 + 273 = 1198 \,\text{K}\) #tag_action#Load Figure 8.30 or other material property charts to get the constants
03

Now, we need to use Figure 8.30 to obtain values for all the constants in the equation. We can assume that the S-590 alloy is a typical material with a given set of creep properties. By referring to Figure 8.30 or other material property charts, we can find the values for material constant \(A\), activation energy \(Q\), and stress exponent \(n\).

Step 3: Convert Diameter to Cross-sectional Area
04

Next, we need to find the cross-sectional area of the cylinder. To do this, we'll use the diameter and the following formula: \(A_c = \frac{\pi d^2}{4}\) Where \(A_c\) is the cross-sectional area, and \(d\) is the diameter. In this case, \(d\) = 12 mm, so: \(A_c = \frac{\pi (12)^2}{4}= \pi（36)≈ 113.1 mm^2 \)

Step 4: Find Creep Life Span and Creep Strain Rate
05

We are given that the component should survive for 500 hours at 925°C. Creep life span (\(t_{creep}\)) is directly related to the initial creep strain rate (\(\dot{\varepsilon}_i\)) and the total creep strain allowed (\(\varepsilon\)) in the material: \(t_{creep} = \frac{\varepsilon}{\dot{\varepsilon}_i} \) Given the requirement that the component should survive for 500 hours, we can find the initial creep strain rate for the given material: \(\dot{\varepsilon}_i = \frac{\varepsilon}{t_{creep}}\) #tag_action#Find or assume an appropriate creep strain (\(\varepsilon\)) for the given material and conditions

Your instructor or references should provide information on the allowable total creep strain (\(\varepsilon\)) for the given material and conditions. Once you have the value for \(\varepsilon\), you can calculate the initial creep strain rate (\(\dot{\varepsilon}_i\)) as: \(\dot{\varepsilon}_i = \frac{\varepsilon}{500 h}\)
06

Calculate Applied Stress

Now that we have values for temperature, constants, and the initial creep strain rate, we can use the equation mentioned in step 1 to find the applied stress: \( \dot{\varepsilon}_i = A\sigma^n e^{\frac{-Q}{RT}} \) Solving for \(\sigma\), we get: \(\sigma = \left(\frac{\dot{\varepsilon}_i}{Ae^{\frac{-Q}{RT}}}\right)^\frac{1}{n}\) Now, you can plug in the values for \(\dot{\varepsilon}_i\), \(A\), \(n\), \(Q\), \(R\), and \(T\) to calculate the applied stress.
07

Calculate Maximum Load

Finally, to find the maximum load (\(P_{max}\)) that can be applied to the cylindrical component for it to survive 500 hours at 925°C, we use the following equation: \(P_{max} = \sigma A_c\) Plug in the value of \(\sigma\) obtained in step 5 and the cross-sectional area \(A_c\) found in step 3 to find the maximum load, \(P_{max}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

S-590 Alloy Properties
Understanding the S-590 alloy is crucial when designing components intended to withstand high temperatures and stresses such as in aerospace and power generation applications.

The S-590 alloy belongs to a group of materials known for their strength and ability to resist deformation at elevated temperatures. The key attributes of the S-590 alloy include high-temperature strength, resistance to thermal fatigue, good ductility, and excellent creep resistance. Its composition typically consists of nickel, chromium, and other alloying elements that contribute to its high performance under stress.

In practice, this means that components made of S-590 alloy, such as turbine blades or engine parts, can maintain their structural integrity for long periods even when exposed to extreme temperatures and significant mechanical loads. It also implies a lower creep strain rate for this material under a given set of circumstances, compared to non-high-temperature materials.
Stress and Temperature Relationship
The performance of materials under load is significantly influenced by the environmental temperature. In the case of the S-590 alloy, like many others, there is a direct relationship between the applied stress and the temperature at which the material is used.

As the temperature increases, the strength of the material tends to decrease. This is due to the increased atomic activity at higher temperatures, which can lead to atomic migrations and eventually result in permanent deformations under stress, known as creep. The creep strain rate equation helps predict this behavior by linking stress (\f\(\f\)), temperature (T), and creep rate (\f\(\f\)) through material-specific constants.

Creep Strain Rate Equation

This equation is highly sensitive to changes in temperature, reflecting the fact that even a small increase in temperature can lead to a significant increase in the creep strain rate, hence a reduction in the material's lifetime under load. By considering this relationship, engineers can ensure that structures are not overburdened and predict when maintenance or replacement might be necessary.
Creep Life Span Calculation
The calculation of a material's creep life span involves understanding how long a component can withstand applied stresses before failing due to creep. This is particularly important for industries where failure can have catastrophic consequences.

To calculate the creep life span of a component made from S-590 alloy, you need to know the creep strain rate, which is directly linked to the total allowable creep strain (\f\(\f\)) over the life of the component. The initial creep strain rate (\f\(\f_i\)) is essential to determine the time it takes for the material to reach its creep strain limit under a specific load.

Importance of Creep Strain Allowance

Careful consideration must be given to the allowable total creep strain for the material, as this dictates the longevity of the component. Materials with higher allowance typically can serve longer in high-stress, high-temperature situations. In the example given, by defining a suitable total creep strain and running the appropriate calculations, you can determine the stress the material can handle and consequently the maximum permissible load for a given time period at a specified temperature.

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Most popular questions from this chapter

Steady-state creep rate data are given in the following table for nickel at \(1000^{\circ} \mathrm{C}(1273 \mathrm{~K})\) : $$ \begin{array}{cc} \hline \dot{\epsilon}_{s}\left(\boldsymbol{s}^{-1}\right) & \boldsymbol{\sigma}[\boldsymbol{M P a}(\boldsymbol{p s i})] \\ \hline 10^{-4} & 15(2175) \\ 10^{-6} & 4.5(650) \\ \hline \end{array} $$ If it is known that the activation energy for creep is \(272,000 \mathrm{~J} / \mathrm{mol}\), compute the steady-state creep rate at a temperature of \(850^{\circ} \mathrm{C}\) (1123 K) and a stress level of \(25 \mathrm{MPa}\) (3625 psi).

An aircraft component is fabricated from an aluminum alloy that has a plane strain fracture toughness of \(35 \mathrm{MPa} \sqrt{\mathrm{m}}(31.9 \mathrm{ksi} \sqrt{\text { in. }})\). It has been determined that fracture results at a stress of \(250 \mathrm{MPa}\) (36,250 psi) when the maximum (or critical) internal crack length is \(2.0 \mathrm{~mm}\) (0.08 in.). For this same component and alloy, will fracture occur at a stress level of \(325 \mathrm{MPa}\) (47,125 psi) when the maximum internal crack length is \(1.0 \mathrm{~mm}(0.04\) in.)? Why or why not?

Suppose that a wing component on an aircraft is fabricated from an aluminum alloy that has a plane strain fracture toughness of \(40 \mathrm{MPa} \sqrt{\mathrm{m}}(36.4 \mathrm{ksi} \sqrt{\mathrm{in}}\).). It has been determined that fracture results at a stress of 365 MPa (53,000 psi) when the maximum internal crack length is \(2.5 \mathrm{~mm}(0.10 \mathrm{in} .)\). For this same component and alloy, compute the stress level at which fracture will occur for a critical internal crack length of \(4.0 \mathrm{~mm}(0.16 \mathrm{in}\).).

A fatigue test was conducted in which the mean stress was \(50 \mathrm{MPa}\) (7250 psi) and the stress amplitude was \(225 \mathrm{MPa}(32,625 \mathrm{psi})\). (a) Compute the maximum and minimum stress levels. (b) Compute the stress ratio. (c) Compute the magnitude of the stress range.

Cite five factors that may lead to scatter in fatigue life data.
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