Subtopic: Brayton Cycle with Real Component Efficiencies
Question Statement:
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 [σ]. 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.
Options:
(A) [Q] [σ] [Q]T
(B) [Q] [σ] [Q]-1
(C) ([Q]-1)T [σ] [Q]T
(D) [Q]-1 [σ] [Q]
Official Answer Key:
A, B, C
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2026 Q.26)**
**1. Intuitive Physical Setup & Given Parameters**:
Let $[\sigma]$ denote the Cauchy stress tensor in the original Cartesian coordinate system $x$-$y$.
Let $[Q]$ denote the coordinate transformation (direction cosine) matrix that transforms vectors as:
$\{x'\} = [Q]\{x\}$
Since coordinate transformations between Cartesian frames are orthogonal, $[Q]$ satisfies:
$[Q]^{-1} = [Q]^T$
We need to determine the correct matrix representation of the transformed stress tensor $[\sigma']$ in the new coordinate system $x'$-$y'$ (MSQ).
**2. Step-by-Step Tensor Transformation Derivation**:
1. **Tensor Transformation Rule**:
- Stress is a second-order Cartesian tensor. Under a coordinate transformation with direction cosines $Q_{ij}$, the transformation law in indicial notation is:
$\sigma'_{ij} = Q_{ik} Q_{jl} \sigma_{kl} = Q_{ik} \sigma_{kl} (Q^T)_{lj}$
- In matrix notation, this corresponds to:
$[\sigma'] = [Q] [\sigma] [Q]^T$
- Thus, **Option A** is correct.
2. **Utilizing Orthogonality of $[Q]$**:
- Because $[Q]$ represents a pure Cartesian coordinate rotation, it is an orthogonal matrix:
$[Q]^T = [Q]^{-1}$
- Substituting $[Q]^T = [Q]^{-1}$ into Option A yields:
$[\sigma'] = [Q] [\sigma] [Q]^{-1}$
- Thus, **Option B** is also correct.
3. **Checking Inverse Transpose**:
- Since $[Q]^{-1} = [Q]^T$, taking transpose on both sides gives $([Q]^{-1})^T = Q$.
- Therefore:
$([Q]^{-1})^T [\sigma] [Q]^T = [Q] [\sigma] [Q]^T = [\sigma']$
- Thus, **Option C** is also mathematically equivalent and correct.
4. **Checking Option D**:
- Option D is $[Q]^{-1} [\sigma] [Q] = [Q]^T [\sigma] [Q]$, which represents the inverse transformation from $x'$ back to $x$, not from $x$ to $x'$. Thus, Option D is incorrect.
**3. Comprehensive Options & Distractor Analysis**:
- **(A) [Q] [σ] [Q]T**: ✅ **CORRECT**. Standard second-order tensor transformation formula.
- **(B) [Q] [σ] [Q]-1**: ✅ **CORRECT**. Equivalent to A because $[Q]^T = [Q]^{-1}$ for orthogonal Cartesian rotations.
- **(C) ([Q]-1)T [σ] [Q]T**: ✅ **CORRECT**. Since $([Q]^{-1})^T = ([Q]^T)^T = [Q]$, this evaluates identically to $[Q][\sigma][Q]^T$.
- **(D) [Q]-1 [σ] [Q]**: ❌ **INCORRECT**. This represents the backwards transformation from the primed system to the unprimed system.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **A, B, C**
- **Exam Takeaway**: For orthogonal Cartesian coordinate rotations, always remember $[Q]^T = [Q]^{-1}$. Both $[Q][\sigma][Q]^T$ and $[Q][\sigma][Q]^{-1}$ represent the exact same forward tensor rotation.
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