3.62 Given that√3 1 0[F] = 0 20001determine the right and left stretch tensors.

Answered: 3.62 Given that √3 1 07 [F] = 0 20 001…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

3.62 Given that
√3 1 0
[F] = 0 20
001
determine the right and left stretch tensors.

Expert Solution

Step 1: Finding the streatch tensor

The right Cauchy-Green tensor $C$ is related to the deformation gradient $F$ by

$C = F^{T} F$

Given $F = [\begin{matrix} \sqrt{3} & 1 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{matrix}]$

$∴ C = [\begin{matrix} \sqrt{3} & 0 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & 1 \end{matrix}] [\begin{matrix} \sqrt{3} & 1 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{matrix}] = [\begin{matrix} 3 & \sqrt{3} & 0 \\ \sqrt{3} & 5 & 0 \\ 0 & 0 & 1 \end{matrix}]$

The left Cauchy-Green tensor $B$ is related to the deformation gradient $F$ by

$B = F F^{T} = [\begin{matrix} \sqrt{3} & 1 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{matrix}] [\begin{matrix} \sqrt{3} & 0 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & 1 \end{matrix}] = [\begin{matrix} 4 & 2 & 0 \\ 2 & 4 & 0 \\ 0 & 0 & 1 \end{matrix}]$