Learning Goal: A material is homogenous and isotropic when it has a modulus of elasticity that does not vary with direction and it has uniform properties throughout. When a point in a material is subjected to only normal stresses in three dimensions, but they are not necessarily equal, the state is called triaxial stress. When the response of the material is linear elastic, the strains along the three axes can be calculated using a generalized form of Hooke's law Each strain depends on all three stresses because of the Poisson effect - Flo, v (a₂ + a₂)] - ty=. Figure < 1 of 1 A cube is subjected to stresses with magnitudes a = 16 MPa, b= 26 MPa, and c= 5 MPa (Eigure 1) What is the strain in the y-direction? Let = 200 GPa and = 0.26 Express your answer to three significant figures. View Available Hint(s) ▸ €y = Submit VAXI vec ▼ Part B - Change in volume ▸ View Available Hint(s) The cube from Part A originally has side length 3.2 cm. What is the change in volume of the cube under the given stresses? Express your answer with appropriate units to three significant figures. change in volume = 4 Value @ 5 d ? Units ? C
Learning Goal: A material is homogenous and isotropic when it has a modulus of elasticity that does not vary with direction and it has uniform properties throughout. When a point in a material is subjected to only normal stresses in three dimensions, but they are not necessarily equal, the state is called triaxial stress. When the response of the material is linear elastic, the strains along the three axes can be calculated using a generalized form of Hooke's law Each strain depends on all three stresses because of the Poisson effect - Flo, v (a₂ + a₂)] - ty=. Figure < 1 of 1 A cube is subjected to stresses with magnitudes a = 16 MPa, b= 26 MPa, and c= 5 MPa (Eigure 1) What is the strain in the y-direction? Let = 200 GPa and = 0.26 Express your answer to three significant figures. View Available Hint(s) ▸ €y = Submit VAXI vec ▼ Part B - Change in volume ▸ View Available Hint(s) The cube from Part A originally has side length 3.2 cm. What is the change in volume of the cube under the given stresses? Express your answer with appropriate units to three significant figures. change in volume = 4 Value @ 5 d ? Units ? C
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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![**Educational Website Transcription**
---
**Learning Goal:**
A material is homogeneous and isotropic when it has a modulus of elasticity that does not vary with direction and has uniform properties throughout. When a point in a material is subjected to only normal stresses in three dimensions, but they are not necessarily equal, the state is called triaxial stress. When the response of the material is linear elastic, the strains along the three axes can be calculated using a generalized form of Hooke’s Law. Each strain depends on all three stresses because of the Poisson effect.
\[
\varepsilon_x = \frac{1}{E} [\sigma_x - \nu (\sigma_y + \sigma_z)]
\]
**Diagram Explanation:**
The diagram (Figure 1) illustrates a cube subjected to triaxial stress, with axes labeled \(x\), \(y\), and \(z\). Each axis has an arrow indicating the direction of the stress, with the cube's orientation centered at the origin of the coordinate system.
**Problem Statement:**
A cube is subjected to stresses with magnitudes \(a = 16 \, \text{MPa}\), \(b = 26 \, \text{MPa}\), and \(c = 5 \, \text{MPa}\). What is the strain in the \(y\)-direction? Let \(E = 200 \, \text{GPa}\) and \(\nu = 0.26\).
- **Express your answer to three significant figures.**
\[
\varepsilon_y =
\]
**Submit Button**
---
**Part B - Change in Volume**
The cube from Part A originally has a side length of 3.2 cm. What is the change in volume of the cube under the given stresses?
- **Express your answer with appropriate units to three significant figures.**
\[
\text{change in volume} =
\]
**Submit Button**
---](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffe15d827-124c-48d6-8956-c03935276d05%2Ff30988c8-3a98-46ab-abe4-a1edf861ca27%2Fxks7mzi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Educational Website Transcription**
---
**Learning Goal:**
A material is homogeneous and isotropic when it has a modulus of elasticity that does not vary with direction and has uniform properties throughout. When a point in a material is subjected to only normal stresses in three dimensions, but they are not necessarily equal, the state is called triaxial stress. When the response of the material is linear elastic, the strains along the three axes can be calculated using a generalized form of Hooke’s Law. Each strain depends on all three stresses because of the Poisson effect.
\[
\varepsilon_x = \frac{1}{E} [\sigma_x - \nu (\sigma_y + \sigma_z)]
\]
**Diagram Explanation:**
The diagram (Figure 1) illustrates a cube subjected to triaxial stress, with axes labeled \(x\), \(y\), and \(z\). Each axis has an arrow indicating the direction of the stress, with the cube's orientation centered at the origin of the coordinate system.
**Problem Statement:**
A cube is subjected to stresses with magnitudes \(a = 16 \, \text{MPa}\), \(b = 26 \, \text{MPa}\), and \(c = 5 \, \text{MPa}\). What is the strain in the \(y\)-direction? Let \(E = 200 \, \text{GPa}\) and \(\nu = 0.26\).
- **Express your answer to three significant figures.**
\[
\varepsilon_y =
\]
**Submit Button**
---
**Part B - Change in Volume**
The cube from Part A originally has a side length of 3.2 cm. What is the change in volume of the cube under the given stresses?
- **Express your answer with appropriate units to three significant figures.**
\[
\text{change in volume} =
\]
**Submit Button**
---
![A point in a material experiences strain \( \varepsilon_z = 200 \times 10^{-6} \). The strains along the other two axes are zero. What is the stress in the z-direction? Use \( E = 200 \, \text{GPa} \) and \( \nu = 0.26 \).
Express your answer with appropriate units to three significant figures.
[View Available Hint(s)]
\[
\sigma_z = \begin{array}{c} \text{Value} \\ \text{Units} \end{array}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffe15d827-124c-48d6-8956-c03935276d05%2Ff30988c8-3a98-46ab-abe4-a1edf861ca27%2Fcey0w4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:A point in a material experiences strain \( \varepsilon_z = 200 \times 10^{-6} \). The strains along the other two axes are zero. What is the stress in the z-direction? Use \( E = 200 \, \text{GPa} \) and \( \nu = 0.26 \).
Express your answer with appropriate units to three significant figures.
[View Available Hint(s)]
\[
\sigma_z = \begin{array}{c} \text{Value} \\ \text{Units} \end{array}
\]
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