Given the following state of stress at a point in a continuum, 0 0 Ar₂ 0 0 - Br3 MPa, Ar₂ -Bra 0 [0] = where A and B are constants. Determine (a) the body force vector such that the stress tensor corresponds to an equilibrium state, (b) the three principal invariants of at the point x = Bê₂ + Aê3, (e) the principal stress components and the associated planes at the point x = Bê₂ + Aê3, and (d) the maximum shear stress and associated plane at the point x = Bê₂ + Aê3.

Given the following state of stress at a point in a continuum, 0 0 Ar₂ 0 0 - Br3 MPa, Ar₂ -Bra 0 [0] = where A and B are constants. Determine (a) the body force vector such that the stress tensor corresponds to an equilibrium state, (b) the three principal invariants of at the point x = Bê₂ + Aê3, (e) the principal stress components and the associated planes at the point x = Bê₂ + Aê3, and (d) the maximum shear stress and associated plane at the point x = Bê₂ + Aê3.

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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Pressure Vessels

The term pressure vessel refers to a closed container that contains gas or liquid at a different pressure than environment pressure. The pressure vessels have a wide scale of applications in our day-to-day lives and other industrial places. Some common examples of pressure vessels are gas cylinders, autoclaves, compressor vessels, etc.

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**4.27** Given the following state of stress at a point in a continuum,

\[
[\sigma] = \begin{bmatrix}
0 & 0 & Ax_2 \\
0 & 0 & -Bx_3 \\
Ax_2 & -Bx_3 & 0
\end{bmatrix}
\text{ MPa},
\]

where \( A \) and \( B \) are constants. Determine:

(a) the body force vector such that the stress tensor corresponds to an equilibrium state,

(b) the three principal invariants of \(\sigma\) at the point \( x = B\hat{e}_2 + A\hat{e}_3 \),

(c) the principal stress components and the associated planes at the point \( x = B\hat{e}_2 + A\hat{e}_3 \), and

(d) the maximum shear stress and associated plane at the point \( x = B\hat{e}_2 + A\hat{e}_3 \).

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