--- title: "Volumetric strain is three times longitudinal strain (cube)" subject: "Strength of Materials" chapter: 3 key: "semester_3/strength_of_materials:volumetric-strain-three-times-longitudinal" source: https://www.bctengineer.com/q/semester_3/strength_of_materials/volumetric-strain-three-times-longitudinal site: BCT Engineering --- # Volumetric strain is three times longitudinal strain (cube) Strength of Materials · Chapter 3: Simple Stress and Strain - Asked in: 2066 Jestha (old course), 8 marks - Video: [Problem on Volumetric Strain of rectangular bar (BRB)](https://www.youtube.com/watch?v=LW8QrMxbTDw) — Sampurna Engineering - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/strength_of_materials/volumetric-strain-three-times-longitudinal - Whole chapter: https://www.bctengineer.com/md/semester_3/strength_of_materials/chapter-3 ## Question Prove that the volumetric strain is three times the longitudinal strain for a cube subjected to equal stresses in the three mutually perpendicular directions. ## Short answer For a cube of side L under equal triaxial stress, new volume V' = L³(1+ε)³ ≈ V(1+3ε) for small ε. Thus volumetric strain δV/V ≈ 3ε, proving it is three times the longitudinal strain. ## Answer **Statement:** for a cube under equal stress $\sigma$ in three mutually perpendicular directions (the same type of stress on all faces), the volumetric strain $\varepsilon_v = 3\varepsilon$, where $\varepsilon$ is the linear strain along any edge. **Proof.** Let the cube have side $L$, so $V = L^3$. Under the stress, every edge changes by the same amount to $L + \delta L$, giving the linear strain $$ \varepsilon = \frac{\delta L}{L} $$ New volume: $$ V' = (L+\delta L)^3 = L^3\left(1+\varepsilon\right)^3 = V\left(1 + 3\varepsilon + 3\varepsilon^2 + \varepsilon^3\right) $$ Change in volume: $$ \delta V = V' - V = V\left(3\varepsilon + 3\varepsilon^2 + \varepsilon^3\right) $$ As the strains are very small, $\varepsilon^2$ and $\varepsilon^3$ are negligible compared with $\varepsilon$: $$ \varepsilon_v = \frac{\delta V}{V} \approx 3\varepsilon $$ **Alternative (differentiation):** from $V = L^3$, $dV = 3L^2\,dL$, so $\dfrac{dV}{V} = 3\dfrac{dL}{L}$, which gives the same result. **Note (with Poisson's effect):** the strain along one edge caused by the three equal stresses is $\varepsilon = \dfrac{\sigma}{E} - \dfrac{\nu\sigma}{E} - \dfrac{\nu\sigma}{E} = \dfrac{\sigma}{E}(1-2\nu)$, so $$ \varepsilon_v = 3\varepsilon = \frac{3\sigma}{E}(1-2\nu) = \frac{\sigma}{K} $$ since $K = \dfrac{E}{3(1-2\nu)}$. Hence $\varepsilon_v = 3\varepsilon$ is proved. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is the starting relation used in the proof? A. V = L^3 and the new volume V' = (L + dL)^3 B. V = 6L^2 C. V = L^2 and V' = (L + dL)^2 D. V = 3L Answer: A. The cube volume is L^3, and every edge changes by the same dL. (Key point: Volume of cube) 2. Why are the eps^2 and eps^3 terms dropped? A. They are exactly zero for a cube B. They cancel with Poisson's effect C. Strains are very small, so they are negligible compared with eps D. They represent shear strain Answer: C. With eps tiny, eps^2 and eps^3 are far smaller than 3 eps, giving dV/V = 3 eps. (Key point: Neglecting higher terms) 3. Which differentiation gives the same result? A. dV = 3L^2 dL, so dV/V = 3 dL/L B. dV = L^2 dL, so dV/V = dL/L C. dV = 6L dL, so dV/V = 6 dL/L D. dV = 3L dL, so dV/V = 3 dL/L^2 Answer: A. Differentiating V = L^3 gives dV = 3L^2 dL; dividing by V gives 3 dL/L. (Key point: Alternative by differentiation) 4. With equal stress sigma on all three faces, what is the edge strain? A. sigma/E B. (sigma/E)(1 + 2nu) C. (sigma/E)(1 - 2nu) D. (sigma/E)(1 - nu) Answer: C. Each stress gives sigma/E along its own direction and -nu sigma/E from the other two, so eps = (sigma/E)(1-2nu) and eps_v = 3 eps = sigma/K. (Key point: Poisson's effect and K) ## Related in Strength of Materials - [Volumetric strain of thin cylindrical vessel](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Avolumetric-strain-thin-cylinder-derive) — Strength of Materials, chapter 5, asked 2× - [Max shear stress and volumetric strain in thin cylinder](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Amax-shear-half-longitudinal-volumetric-strain) — Strength of Materials, chapter 5 - [Stresses on 20 degree plane; principal and max shear](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Ainclined-plane-20deg-50-40-80) — Strength of Materials, chapter 4, asked 4× - [Normal/shear stress on 25 degree plane, principal stresses](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Ainclined-plane-30-25deg-mohr-verify) — Strength of Materials, chapter 4, asked 3× - [Stresses on 30 degree plane; Mohr's circle](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Ainclined-plane-30deg-50-30-mohr) — Strength of Materials, chapter 4, asked 3× - [Longitudinal stress is half of circumferential stress](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Alongitudinal-stress-half-circumferential) — Strength of Materials, chapter 5, asked 2× ## Related in other subjects - [Prove q = k times focal length](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Adischarge-equals-k-focal-length) — Soil Mechanics, chapter 7 - [Strain energy density and thermal stress](https://www.bctengineer.com/question.md?key=semester_5%2Fmechanics_of_solids%3Astrain-energy-density-thermal-constraint) — Mechanics of Solids, chapter 4 - [Strain rosette: principal strains and stresses](https://www.bctengineer.com/question.md?key=semester_5%2Fmechanics_of_solids%3Astrain-rosette-principal-numerical) — Mechanics of Solids, chapter 3 - [Best of three mutually exclusive projects, 10-year study](https://www.bctengineer.com/question.md?key=semester_6%2Fengineering_economics%3Amutually-exclusive-abc-study-10-years) — Engineering Economics, chapter 4