---
title: "Material response, isotropy and temperature effect"
subject: "Mechanics of Solids"
chapter: 1
key: "semester_5/mechanics_of_solids:elastic-response-isotropy-temperature"
source: https://www.bctengineer.com/q/semester_5/mechanics_of_solids/elastic-response-isotropy-temperature
site: BCT Engineering
---

# Material response, isotropy and temperature effect

Mechanics of Solids · Chapter 1: Load on Structure and Response of Material

- Asked in: year not given, 5 marks
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/mechanics_of_solids/elastic-response-isotropy-temperature
- Whole chapter: https://www.bctengineer.com/md/semester_5/mechanics_of_solids/chapter-1

## Question

Differentiate between (a) elastic and non-elastic response of a solid, (b) isotropic and anisotropic materials, and (c) a homogeneous and a continuous body. State how temperature affects the elastic and plastic range of a solid.

## Answer

#### (a) Elastic and non-elastic response

| Point | Elastic | Non-elastic (plastic, viscoelastic) |
|---|---|---|
| After unloading | Body returns fully to original shape | Some permanent (residual) strain remains |
| Stress-strain | Loading and unloading follow the same path | Unloading path differs; energy is dissipated |
| Region | Up to the elastic limit (yield) | Beyond yield |
| Example | Steel spring below yield | Steel after necking; clay; lead |

Linear elastic response obeys Hooke's law, $\sigma = E\varepsilon$.

#### (b) Isotropic and anisotropic

| Point | Isotropic | Anisotropic |
|---|---|---|
| Properties | Same in all directions at a point | Depend on direction |
| Independent elastic constants | 2 ($E$, $\nu$) | Up to 21 (9 for orthotropic) |
| Example | Mild steel, aluminium (ideal) | Timber, fibre composites, rolled sheet |

#### (c) Homogeneous and continuous

- **Homogeneous**: properties are the same at every point of the body (they do not change with position). A concrete block with voids and variable aggregate is not exactly homogeneous.
- **Continuous**: the body has no gaps or voids, so matter is distributed without break and quantities such as stress, strain and displacement are continuous functions of position. This allows calculus to be used.

A body can be homogeneous but anisotropic (a straight timber log of uniform fibre direction).

#### Effect of temperature

- As temperature rises, $E$ and yield strength fall; the **elastic range shortens** and the **plastic range** (ductility) usually **increases**. Mild steel loses most of its strength near 500 to 600 $^\circ$C.
- At low temperature, yield strength rises but ductility falls, so steel can become brittle (ductile-brittle transition).
- At high temperature under steady load, materials creep (strain increases with time).
- If expansion is restrained, a thermal stress $\sigma = E\alpha\Delta T$ develops.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What distinguishes non-elastic response from elastic response?
   A. Loading and unloading follow the same path
   B. Permanent strain remains after unloading
   C. Hooke's law holds exactly
   D. It occurs only below the elastic limit
   Answer: B. In non-elastic response unloading follows a different path and residual strain remains, with energy dissipated. (Key point: Elastic vs non-elastic)

2. How many independent elastic constants does an isotropic material have?
   A. 21
   B. 9
   C. 5
   D. 2
   Answer: D. An isotropic material needs only two independent constants, E and nu. (Key point: Isotropic constants)

3. Which statement correctly describes a continuous body?
   A. It has no gaps or voids, so stress and strain are continuous functions of position
   B. Its properties are the same in every direction
   C. Its properties are the same at every point
   D. It always returns to its original shape
   Answer: A. Continuity means no voids, which allows calculus to be used; sameness of properties at all points is homogeneity. (Key point: Homogeneous vs continuous)

4. How does a rise in temperature affect the elastic and plastic range of steel?
   A. Elastic range lengthens and ductility falls
   B. Both ranges stay unchanged
   C. Elastic range shortens and ductility usually increases
   D. Yield strength rises
   Answer: C. E and yield strength fall with temperature, shortening the elastic range while the plastic range usually increases. (Key point: Effect of temperature)

## Related in Mechanics of Solids

- [Generalised Hooke's law for isotropic materials](https://www.bctengineer.com/question.md?key=semester_5%2Fmechanics_of_solids%3Ageneralised-hooke-law-isotropic) — Mechanics of Solids, chapter 4
- [Elastic-plastic torsion of a circular shaft](https://www.bctengineer.com/question.md?key=semester_5%2Fmechanics_of_solids%3Aplastic-torsion-circular-shaft) — Mechanics of Solids, chapter 12
- [Plastic moment and shape factor of a beam](https://www.bctengineer.com/question.md?key=semester_5%2Fmechanics_of_solids%3Ashape-factor-plastic-moment-rectangle) — Mechanics of Solids, chapter 12

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