Question

The state of stress at a point in a 2-D body, in the \(x\)-\(y\) Cartesian coordinate system, is represented in matrix form as \([\sigma]\). The transformation matrix \([Q]\) rotates the coordinate system to a new \(x'\)-\(y'\) Cartesian coordinate system. Select the CORRECT option(s) that represent(s) the state of stress in the new coordinate system.

• A. \([Q][\sigma][Q]^{T}\)
• B. \([Q][\sigma][Q]^{-1}\)
• C. \(([Q]^{-1})^{T}[\sigma][Q]^{T}\)
• D. \([Q]^{-1}[\sigma][Q]\)

Hint:

Stress transforms as \([\sigma']=[Q][\sigma][Q]^{T}\); since \([Q]\) is orthogonal, \([Q]^{-1}=[Q]^{T}\), so any option that algebraically reduces to this form is also correct.

Answer 1

The Correct Option is A, B, C

Step 1: Understanding the Concept.
Stress is a second rank tensor, so under a coordinate rotation its components transform using the direction cosine matrix twice, once for each index: in index notation, $\sigma'_{ij} = Q_{ik}Q_{jl}\sigma_{kl}$. Written in matrix form this is $[\sigma'] = [Q][\sigma][Q]^T$.

Step 2: Key Formula or Approach.
The one fact that decides this whole question is that $[Q]$, being a rotation matrix connecting two Cartesian frames, is orthogonal: $[Q]^T[Q] = [Q][Q]^T = [I]$, so $[Q]^{-1} = [Q]^T$. Every option should be simplified using this single substitution and then compared against the reference formula $[Q][\sigma][Q]^T$.

Step 3: Detailed Explanation.
Option (A) is already the reference formula, so it is correct. Option (B), $[Q][\sigma][Q]^{-1}$, becomes $[Q][\sigma][Q]^{T}$ once $[Q]^{-1}$ is replaced by $[Q]^{T}$, matching the reference exactly, so it is correct. Option (C), $([Q]^{-1})^{T}[\sigma][Q]^{T}$, first replace $[Q]^{-1}$ with $[Q]^T$ to get $([Q]^{T})^{T}[\sigma][Q]^{T}$, and since transposing twice returns the original matrix, $([Q]^{T})^{T} = [Q]$, this becomes $[Q][\sigma][Q]^{T}$ again, matching the reference, so it is also correct. Option (D), $[Q]^{-1}[\sigma][Q]$, becomes $[Q]^{T}[\sigma][Q]$ after the substitution. Here the transpose sits on the left factor and the plain $[Q]$ sits on the right, the reverse arrangement compared to the reference formula $[Q][\sigma][Q]^T$; these two expressions are equal only if $[Q]$ is symmetric, which is not true for a general rotation, so option (D) fails.

Step 4: Final Answer.
Options (A), (B) and (C) all represent the correctly transformed stress matrix; option (D) does not.