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 G = 1 Roz-v(o₂ +0₂)]
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 G = 1 Roz-v(o₂ +0₂)]
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
Related questions
Question
Parts a and b were answered in a previous question, part c was unanswered this is the entire question :)
![---
**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.
\[
\varepsilon_x = \frac{1}{E} [\sigma_x - \nu (\sigma_y + \sigma_z)]
\]
**Figure:**
---
**Problem Statement:**
A cube is subjected to stresses with magnitudes \(a = 16 \, \text{MPa}, \, b = 26 \, \text{MPa},\) and \(c = 5 \, \text{MPa} \, (\text{Figure 1}).\) What is the strain in the \(y\)-direction? Let \(E = 200 \, \text{GPa}\) and \(\nu = 0.26\).
**Express your answer to three significant figures.**
*Strain Calculation Form:*
\[
\varepsilon_y =
\]
**Part B - Change in Volume:**
The cube from Part A originally has a side length of \(3.2 \, \text{cm}\). What is the change in volume of the cube under the given stresses?
**Express your answer with appropriate units to three significant figures.**
*Volume Change Calculation Form:*
\[
\text{change in volume} = \, \text{Value} \, \text{Units}
\]
---](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffe15d827-124c-48d6-8956-c03935276d05%2Fd42740b9-d072-44dc-9dbc-a8884f4e5d4e%2Fvxtvnen_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
**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.
\[
\varepsilon_x = \frac{1}{E} [\sigma_x - \nu (\sigma_y + \sigma_z)]
\]
**Figure:**
---
**Problem Statement:**
A cube is subjected to stresses with magnitudes \(a = 16 \, \text{MPa}, \, b = 26 \, \text{MPa},\) and \(c = 5 \, \text{MPa} \, (\text{Figure 1}).\) What is the strain in the \(y\)-direction? Let \(E = 200 \, \text{GPa}\) and \(\nu = 0.26\).
**Express your answer to three significant figures.**
*Strain Calculation Form:*
\[
\varepsilon_y =
\]
**Part B - Change in Volume:**
The cube from Part A originally has a side length of \(3.2 \, \text{cm}\). What is the change in volume of the cube under the given stresses?
**Express your answer with appropriate units to three significant figures.**
*Volume Change Calculation Form:*
\[
\text{change in volume} = \, \text{Value} \, \text{Units}
\]
---
![**Educational Text with Explanation**
**Learning Goal:**
A material is homogeneous 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.
\[ \varepsilon_x = \frac{1}{E} (\sigma_x - \nu (\sigma_y + \sigma_z)) \]
**Figure:**
A 3D coordinate system is shown with an axis-aligned cube. The axes are labeled \( x \), \( y \), and \( z \), and each axis has an arrow pointing in the positive direction. The cube represents a material subjected to triaxial stress.
**Part C - Calculate Stress:**
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.
---
A user interface is present for inputting the stress \( \sigma_z \) in megapascals (MPa). The box has the value "9.458" MPa entered, and there is a "Submit" button below. An error message states:
"Incorrect; Try Again; 5 attempts remaining."
Additional options for viewing available hints, previous answers, and providing feedback are also visible.
**Instructions:**
1. Calculate the stress in the z-direction using the given strain in the z-direction and the material properties provided.
2. Ensure the answer has the appropriate units (MPa) and is rounded to three significant figures before submission.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffe15d827-124c-48d6-8956-c03935276d05%2Fd42740b9-d072-44dc-9dbc-a8884f4e5d4e%2Fwsktnh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Educational Text with Explanation**
**Learning Goal:**
A material is homogeneous 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.
\[ \varepsilon_x = \frac{1}{E} (\sigma_x - \nu (\sigma_y + \sigma_z)) \]
**Figure:**
A 3D coordinate system is shown with an axis-aligned cube. The axes are labeled \( x \), \( y \), and \( z \), and each axis has an arrow pointing in the positive direction. The cube represents a material subjected to triaxial stress.
**Part C - Calculate Stress:**
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.
---
A user interface is present for inputting the stress \( \sigma_z \) in megapascals (MPa). The box has the value "9.458" MPa entered, and there is a "Submit" button below. An error message states:
"Incorrect; Try Again; 5 attempts remaining."
Additional options for viewing available hints, previous answers, and providing feedback are also visible.
**Instructions:**
1. Calculate the stress in the z-direction using the given strain in the z-direction and the material properties provided.
2. Ensure the answer has the appropriate units (MPa) and is rounded to three significant figures before submission.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps with 3 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.Recommended textbooks for you
Elements Of Electromagnetics
Mechanical Engineering
ISBN:
9780190698614
Author:
Sadiku, Matthew N. O.
Publisher:
Oxford University Press
Mechanics of Materials (10th Edition)
Mechanical Engineering
ISBN:
9780134319650
Author:
Russell C. Hibbeler
Publisher:
PEARSON
Thermodynamics: An Engineering Approach
Mechanical Engineering
ISBN:
9781259822674
Author:
Yunus A. Cengel Dr., Michael A. Boles
Publisher:
McGraw-Hill Education
Elements Of Electromagnetics
Mechanical Engineering
ISBN:
9780190698614
Author:
Sadiku, Matthew N. O.
Publisher:
Oxford University Press
Mechanics of Materials (10th Edition)
Mechanical Engineering
ISBN:
9780134319650
Author:
Russell C. Hibbeler
Publisher:
PEARSON
Thermodynamics: An Engineering Approach
Mechanical Engineering
ISBN:
9781259822674
Author:
Yunus A. Cengel Dr., Michael A. Boles
Publisher:
McGraw-Hill Education
Control Systems Engineering
Mechanical Engineering
ISBN:
9781118170519
Author:
Norman S. Nise
Publisher:
WILEY
Mechanics of Materials (MindTap Course List)
Mechanical Engineering
ISBN:
9781337093347
Author:
Barry J. Goodno, James M. Gere
Publisher:
Cengage Learning
Engineering Mechanics: Statics
Mechanical Engineering
ISBN:
9781118807330
Author:
James L. Meriam, L. G. Kraige, J. N. Bolton
Publisher:
WILEY