13.13 Tungsten is being used at half its melting point (Tm ≈ 3,400 °C) and a stress level of 160 MPa. An engineer suggests increasing the grain size by a factor of 4 as an effective means of reducing the creep rate. (a) Do you agree with the engineer? Why? What if the stress level were equal to 1.6 MPa? (b) What is the predicted increase in length of the specimen after 10,000 hours if the initial length is 10 cm? (Hint: Use a Weertman--Ashby map.) 1) Where in the book/slides is this covered? = page # or slide session/number 2) What equations and concepts apply here? Normalized tensile stress, o/G 10-² 10-4 10-6 10-8 0 Tungsten d=10um, p = 4x 1010/cm² Theoretical strength Dislocation glide Coble creep 0.2 0.4 10-10 Dislocation creep /s 10-6 0.6 0.8 Homologous temperature, T/TM -10-2 10-4 10-6 Nabarro creep 1.0 SS ADGb b KBT d Gw~ 160 GPa Dislocation creep mechanism Dislocation glide climb, climb controlled Dislocation glide climb, glide controlled Dissolution of dislocation loops Dislocation climb without glide Dislocation climb by pipe diffusion Diffusional creep mechanisms Vacancy flow through grains Vacancy flow along grain boundaries Interface reaction control TABLE 17.6 Creep Equation Exponents and Diffusion Paths for Various Creep Mechanisms Creep mechanism Р Diffusion path Grain boundary sliding mechanisms Sliding with liquid Sliding without liquid (diffusion control) 0 3 2-3 n 4-5 3 4 σ 3 5 n Lattice Lattice Lattice Lattice Dislocation core Lattice Grain boundary Lattice/grain boundary Liquid Lattice/grain boundary
13.13 Tungsten is being used at half its melting point (Tm ≈ 3,400 °C) and a stress level of 160 MPa. An engineer suggests increasing the grain size by a factor of 4 as an effective means of reducing the creep rate. (a) Do you agree with the engineer? Why? What if the stress level were equal to 1.6 MPa? (b) What is the predicted increase in length of the specimen after 10,000 hours if the initial length is 10 cm? (Hint: Use a Weertman--Ashby map.) 1) Where in the book/slides is this covered? = page # or slide session/number 2) What equations and concepts apply here? Normalized tensile stress, o/G 10-² 10-4 10-6 10-8 0 Tungsten d=10um, p = 4x 1010/cm² Theoretical strength Dislocation glide Coble creep 0.2 0.4 10-10 Dislocation creep /s 10-6 0.6 0.8 Homologous temperature, T/TM -10-2 10-4 10-6 Nabarro creep 1.0 SS ADGb b KBT d Gw~ 160 GPa Dislocation creep mechanism Dislocation glide climb, climb controlled Dislocation glide climb, glide controlled Dissolution of dislocation loops Dislocation climb without glide Dislocation climb by pipe diffusion Diffusional creep mechanisms Vacancy flow through grains Vacancy flow along grain boundaries Interface reaction control TABLE 17.6 Creep Equation Exponents and Diffusion Paths for Various Creep Mechanisms Creep mechanism Р Diffusion path Grain boundary sliding mechanisms Sliding with liquid Sliding without liquid (diffusion control) 0 3 2-3 n 4-5 3 4 σ 3 5 n Lattice Lattice Lattice Lattice Dislocation core Lattice Grain boundary Lattice/grain boundary Liquid Lattice/grain boundary
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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![**Problem Statement:**
13.13 Tungsten is being used at half its melting point (\(T_m \approx 3,400 \, ^\circ C\)) and a stress level of 160 MPa. An engineer suggests increasing the grain size by a factor of 4 as an effective means of reducing the creep rate.
(a) Do you agree with the engineer? Why? What if the stress level were equal to 1.6 MPa?
(b) What is the predicted increase in length of the specimen after 10,000 hours if the initial length is 10 cm?
(Hint: Use a Weertman–Ashby map.)
1) Where in the book/slides is this covered? = page # or slide session/number
2) What equations and concepts apply here?
**Diagram Explanation:**
The graph shows the relationship between normalized tensile stress (\(\sigma/G\)) on the y-axis and homologous temperature (\(T/T_m\)) on the x-axis. The chart indicates different creep regimes for tungsten with a grain size (\(d\)) of 10 μm and a dislocation density (\(\rho\)) of \(4 \times 10^{10}/{\text{cm}}^2\).
- **Coble creep** dominates at low temperatures and stresses.
- **Dislocation creep** begins at moderate temperatures and stresses, transitioning from dislocation glide to climb.
- At high temperatures and stresses, **Nabarro-Herring creep** is dominant.
**Equations and Concepts:**
The steady-state creep rate equation is given by:
\[
\dot{\varepsilon}_{\text{ss}} = \frac{ADGb}{k_BT} \left(\frac{b}{d}\right)^p \left(\frac{\sigma}{G}\right)^n
\]
Where:
- \(A\) is a constant,
- \(D\) is the diffusion coefficient,
- \(G\) is the shear modulus (\(G_W \approx 160 \, \text{GPa}\)),
- \(b\) is the Burgers vector,
- \(d\) is the grain size,
- \(k_B\) is Boltzmann's constant,
- \(T\) is the temperature,
- \(\sigma\) is the applied stress.
**Table Explanation:**
*Table 17.6: Creep Equation Exponents and Diffusion Paths for Various](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3752cebe-44a2-41dc-8aad-9e7c48b99250%2F2b1b577b-ab47-45ff-855e-ae04e41784dd%2F77wk6s_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
13.13 Tungsten is being used at half its melting point (\(T_m \approx 3,400 \, ^\circ C\)) and a stress level of 160 MPa. An engineer suggests increasing the grain size by a factor of 4 as an effective means of reducing the creep rate.
(a) Do you agree with the engineer? Why? What if the stress level were equal to 1.6 MPa?
(b) What is the predicted increase in length of the specimen after 10,000 hours if the initial length is 10 cm?
(Hint: Use a Weertman–Ashby map.)
1) Where in the book/slides is this covered? = page # or slide session/number
2) What equations and concepts apply here?
**Diagram Explanation:**
The graph shows the relationship between normalized tensile stress (\(\sigma/G\)) on the y-axis and homologous temperature (\(T/T_m\)) on the x-axis. The chart indicates different creep regimes for tungsten with a grain size (\(d\)) of 10 μm and a dislocation density (\(\rho\)) of \(4 \times 10^{10}/{\text{cm}}^2\).
- **Coble creep** dominates at low temperatures and stresses.
- **Dislocation creep** begins at moderate temperatures and stresses, transitioning from dislocation glide to climb.
- At high temperatures and stresses, **Nabarro-Herring creep** is dominant.
**Equations and Concepts:**
The steady-state creep rate equation is given by:
\[
\dot{\varepsilon}_{\text{ss}} = \frac{ADGb}{k_BT} \left(\frac{b}{d}\right)^p \left(\frac{\sigma}{G}\right)^n
\]
Where:
- \(A\) is a constant,
- \(D\) is the diffusion coefficient,
- \(G\) is the shear modulus (\(G_W \approx 160 \, \text{GPa}\)),
- \(b\) is the Burgers vector,
- \(d\) is the grain size,
- \(k_B\) is Boltzmann's constant,
- \(T\) is the temperature,
- \(\sigma\) is the applied stress.
**Table Explanation:**
*Table 17.6: Creep Equation Exponents and Diffusion Paths for Various
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