Question

The state of strain at a point in a machine component is given as \[ \varepsilon_{xx} = 2.5 \times 10^{-4}, \varepsilon_{yy} = 2.0 \times 10^{-4}, \varepsilon_{zz} = -1.5 \times 10^{-4}, \] \[ \varepsilon_{xy} = 2.5 \times 10^{-4}, \varepsilon_{yz} = -0.5 \times 10^{-4}, \varepsilon_{zx} = -1.0 \times 10^{-4}. \] The volumetric strain at this point is .............

• A. $4 \times 10^{-4}$
• B. $3 \times 10^{-4}$
• C. $-5 \times 10^{-4}$
• D. $-3 \times 10^{-4}$

Hint:

Volumetric strain = sum of normal strains ($\varepsilon_{xx} + \varepsilon_{yy} + \varepsilon_{zz}$). Shear strains do not contribute to volumetric change, only to distortion.

Answer 1

The Correct Option is B

Step 1: Recall volumetric strain definition.
The volumetric strain is the sum of the normal strains along the three coordinate directions: \[ \varepsilon_{vol} = \varepsilon_{xx} + \varepsilon_{yy} + \varepsilon_{zz} \]
Step 2: Substitute given values.
\[ \varepsilon_{vol} = (2.5 \times 10^{-4}) + (2.0 \times 10^{-4}) + (-1.5 \times 10^{-4}) \]
Step 3: Simplify.
\[ \varepsilon_{vol} = (2.5 + 2.0 - 1.5) \times 10^{-4} = 3.0 \times 10^{-4} \] Wait – check carefully: \[ 2.5 + 2.0 = 4.5, 4.5 - 1.5 = 3.0 \] So: \[ \varepsilon_{vol} = 3.0 \times 10^{-4} \]
Step 4: Clarify confusion.
The volumetric strain only includes **normal strains**, not shear strains. Thus, shear strains ($\varepsilon_{xy}, \varepsilon_{yz}, \varepsilon_{zx}$) are irrelevant here. Final Answer: \[ \boxed{3 \times 10^{-4}} \]