**Stress at a Continuum Body Point**The state of stress at a point in a continuum body is given by the stress tensor:\[\sigma = \begin{bmatrix} X & 1.5 & 0 \\ 1.5 & 1.0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \text{ MPa}\]In this context, $X$ is a non-zero value that needs to be determined. This problem involves a brittle material that fractures along the plane of maximum principal stress, with a critical tensile stress of 5 MPa where fracture occurs.### Questions(a) **Determine Maximum Value of $X$:** What is the maximum value of $X$ that the body can sustain without experiencing fracture?(b) **Orientation of the Fracture Plane:** If $X$ reaches the critical tensile stress value, what would the orientation of the fracture plane be? Illustrate the stress state on a properly oriented element.### Explanation- **Matrix Explanation:** - The matrix represents the state of stress in a continuum body, with each element representing stress in a particular direction or plane. - The principal stresses are to be calculated to determine the maximum stress conditions. - **Problem Context:** - The material is brittle, implying fracture occurs at specific stress levels. - The critical tensile stress for fracture is given as 5 MPa, which guides the calculations for $X$.This problem requires understanding principal stresses, tensor analysis, and fracture mechanics to solve for $X$ and determine the fracture plane's orientation.

Answered: Image /qna-images/answer/13359fd5-db64-4fbb-80de-038178af06c4.jpg

Answered: 3. The state of stress at a point in a…

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

See similar textbooks

Related questions

Concept explainers

Question

**Stress at a Continuum Body Point**

The state of stress at a point in a continuum body is given by the stress tensor:

\[
\sigma = \begin{bmatrix} X & 1.5 & 0 \\ 1.5 & 1.0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \text{ MPa}
\]

In this context, $X$ is a non-zero value that needs to be determined. This problem involves a brittle material that fractures along the plane of maximum principal stress, with a critical tensile stress of 5 MPa where fracture occurs.

### Questions

(a) **Determine Maximum Value of $X$:**
What is the maximum value of $X$ that the body can sustain without experiencing fracture?

(b) **Orientation of the Fracture Plane:**
If $X$ reaches the critical tensile stress value, what would the orientation of the fracture plane be? Illustrate the stress state on a properly oriented element.

### Explanation

- **Matrix Explanation:**
- The matrix represents the state of stress in a continuum body, with each element representing stress in a particular direction or plane.
- The principal stresses are to be calculated to determine the maximum stress conditions.

- **Problem Context:**
- The material is brittle, implying fracture occurs at specific stress levels.
- The critical tensile stress for fracture is given as 5 MPa, which guides the calculations for $X$.

This problem requires understanding principal stresses, tensor analysis, and fracture mechanics to solve for $X$ and determine the fracture plane's orientation.

Expert Solution

Step 1: Determine the given data

$Stress space tensor comma straight sigma equals open square brackets table row straight X cell 1.5 end cell 0 row cell 1.5 end cell cell 1.0 end cell 0 row 0 0 0 end table close square brackets Fracture space occurs space at space Principal space stress space hence comma straight sigma subscript 1 equals 5 space MPa To space determine colon straight a right parenthesis The space maximum space value space of space straight X straight b right parenthesis Orientation space of space plane space$

Step by step

Solved in 4 steps with 5 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

Types of Properties of Engineering Materials

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.

Similar questions