4 Stress & Strain
4.1 Direct Stress and Strain
When a bar is subjected to an axial force \(F\) over cross-sectional area \(A\), the direct stress is:
\[\sigma = \frac{F}{A}\]
The resulting axial strain is the ratio of extension to original length:
\[\varepsilon = \frac{\delta L}{L}\]
4.2 Young’s Modulus (Modulus of Elasticity)
Within the elastic limit, stress and strain are proportional — Hooke’s Law:
\[E = \frac{\sigma}{\varepsilon}\]
This gives the deformation of a bar under axial load:
\[\delta L = \frac{FL}{AE}\]
4.3 Shear Stress and Shear Strain
Shear stress acts tangentially across a plane:
\[\tau = \frac{F}{A}\]
Shear strain \(\gamma\) is the angular deformation (in radians). The Modulus of Rigidity (shear modulus) relates them:
\[G = \frac{\tau}{\gamma}\]
4.4 Poisson’s Ratio
Axial loading causes lateral contraction (or expansion) as well as axial strain. Poisson’s ratio \(\nu\) relates the two:
\[\nu = -\frac{\varepsilon_{\text{lat}}}{\varepsilon_{\text{axial}}}\]
Typical values for metals: \(\nu \approx 0.25\)–\(0.33\).
4.5 Volumetric Strain and Bulk Modulus
Under hydrostatic (equal all-round) stress, the volumetric strain is:
\[\varepsilon_v = \frac{\delta V}{V}\]
The Bulk Modulus \(K\) relates hydrostatic stress to volumetric strain:
\[K = \frac{\sigma}{\varepsilon_v}\]
4.6 Relationship Between Elastic Constants
The three elastic constants \(E\), \(G\), and \(K\) are not independent — they are linked through Poisson’s ratio:
\[E = 2G(1 + \nu)\]
\[E = 3K(1 - 2\nu)\]
4.7 Composite Bars
When two materials are bonded together (e.g. a steel rod inside a brass tube), two equations are needed:
Compatibility — both materials deform by the same amount:
\[\delta L_1 = \delta L_2 \quad \Rightarrow \quad \frac{\sigma_1}{E_1} = \frac{\sigma_2}{E_2}\]
Equilibrium — internal forces sum to the applied load:
\[\sigma_1 A_1 + \sigma_2 A_2 = F\]
Solving these simultaneously gives the stress in each material.
4.8 Thermal Stress
If a bar is prevented from expanding freely when heated, a compressive stress is induced:
\[\sigma = E \alpha \Delta T\]
where:
- \(\alpha\) = coefficient of linear thermal expansion
- \(\Delta T\) = temperature change
For a composite bar with mismatched expansion, the same compatibility and equilibrium approach applies.
4.9 Factor of Safety
The factor of safety relates the failure stress to the permissible working stress:
\[\text{FoS} = \frac{\text{failure stress}}{\text{allowable (working) stress}}\]
A higher FoS reflects greater uncertainty in loading or material properties.
4.10 Summary of Equations
| Concept | Equation |
|---|---|
| Direct stress | \(\sigma = F/A\) |
| Direct strain | \(\varepsilon = \delta L / L\) |
| Young’s modulus | \(E = \sigma / \varepsilon\) |
| Axial deformation | \(\delta L = FL/AE\) |
| Shear modulus | \(G = \tau / \gamma\) |
| Poisson’s ratio | \(\nu = -\varepsilon_{\text{lat}}/\varepsilon_{\text{axial}}\) |
| Bulk modulus | \(K = \sigma / \varepsilon_v\) |
| Elastic constants | \(E = 2G(1+\nu) = 3K(1-2\nu)\) |
| Thermal stress | \(\sigma = E\alpha\Delta T\) |
4.11 Summary
This chapter establishes the language of solid mechanics — stress, strain, and the elastic constants — that underpins all subsequent structural chapters. The key skill is setting up compatibility and equilibrium equations correctly for statically indeterminate problems such as composite bars and thermally loaded members.