Elasticity flash cards

1. 1.Define elasticity of a body.

   Elasticity is the property by which a body regains its original shape and size after the deforming force is removed.

   Hint: Restoring property.

2. 2.What is plasticity?

   Plasticity is the property of a body by which it does not regain its original configuration and retains the deformed shape after the removing force is withdrawn.

   Hint: Opposite of elastic; e.g. putty, clay.

3. 3.Define stress.

   Stress is the internal restoring force per unit area developed in a deformed body. $\text{stress}=\dfrac{F}{A}$; SI unit $N/m^2$ (Pa), dimensions $[ML^{-1}T^{-2}]$.

   Hint: Force per unit area.

4. 4.Define strain and state its unit.

   Strain is the fractional change in configuration (ratio of change in dimension to original dimension). It is dimensionless and has no unit.

   Hint: Change/original.

5. 5.What is longitudinal (tensile/normal) stress?

   Force acting normal (perpendicular) to a surface per unit area, producing a change in length: $\sigma=\dfrac{F}{A}$.

   Hint: Force ⟂ to area, along length.

6. 6.Define longitudinal strain.

   Ratio of change in length to original length: $\text{strain}=\dfrac{\Delta L}{L}$.

   Hint: $\Delta L/L$.

7. 7.Distinguish tensile vs compressive stress.

   Tensile stress increases length (stretching, force pulls outward); compressive stress decreases length (force pushes inward). Both are longitudinal (normal) stresses.

   Hint: Pull vs push.

8. 8.What is volume (bulk/hydraulic) stress?

   Force per unit area applied normally and uniformly over the whole surface of a body (e.g. by a fluid), equal to the pressure change $\Delta P$.

   Hint: Uniform normal pressure all around.

9. 9.Define volume strain.

   Ratio of change in volume to original volume: $\dfrac{\Delta V}{V}$.

   Hint: $\Delta V/V$.

10. 10.What is shear (tangential) stress?

    Force acting tangential (parallel) to a surface per unit area: $\sigma_s=\dfrac{F}{A}$, producing a change in shape.

    Hint: Force ∥ to surface.

11. 11.Define shear strain.

    The angle of deformation $\theta$ (in radians) through which a face is tilted: $\theta=\dfrac{\Delta x}{L}$, where $\Delta x$ is relative displacement and $L$ the perpendicular distance between faces.

    Hint: Angle of tilt.

12. 12.State Hooke's law.

    Within the elastic limit, stress is directly proportional to strain: $\text{stress}\propto\text{strain}$, i.e. $\text{stress}=E\times\text{strain}$, where $E$ is the modulus of elasticity.

    Hint: Valid only up to elastic/proportional limit.

13. 13.What is a modulus of elasticity?

    The ratio of stress to strain within the elastic limit: $E=\dfrac{\text{stress}}{\text{strain}}$. Same unit as stress, $N/m^2$ (Pa).

    Hint: Slope of linear stress–strain region.

14. 14.Name the three main moduli of elasticity.

    Young's modulus $Y$ (length/shape via tension), Bulk modulus $K$ (volume), and Modulus of rigidity/shear modulus $\eta$ (shape).

    Hint: Y, K, η.

15. 15.Define Young's modulus $Y$.

    Ratio of longitudinal stress to longitudinal strain: $Y=\dfrac{F/A}{\Delta L/L}=\dfrac{FL}{A\,\Delta L}$.

    Hint: For solids under tension.

16. 16.Write the expression for elongation of a wire under load.

    $\Delta L=\dfrac{FL}{AY}=\dfrac{MgL}{AY}$.

    Hint: Rearrange $Y=FL/(A\Delta L)$.

17. 17.Define bulk modulus $K$.

    Ratio of volume (hydraulic) stress to volume strain: $K=-\dfrac{\Delta P}{\Delta V/V}=-V\dfrac{\Delta P}{\Delta V}$.

    Hint: Negative sign: volume decreases with pressure.

18. 18.Why is there a negative sign in the bulk modulus formula?

    An increase in pressure ($+\Delta P$) causes a decrease in volume ($-\Delta V$). The minus sign makes $K$ positive.

    Hint: Keeps K > 0.

19. 19.Define compressibility.

    Compressibility is the reciprocal of the bulk modulus: $k=\dfrac{1}{K}$. Unit $m^2/N$ (Pa$^{-1}$).

    Hint: $1/K$; ease of compression.

20. 20.Define the modulus of rigidity (shear modulus) $\eta$.

    Ratio of shear (tangential) stress to shear strain: $\eta=\dfrac{F/A}{\theta}=\dfrac{F}{A\theta}$.

    Hint: Resistance to change of shape.

21. 21.Which modulus is relevant for solids, liquids and gases?

    Bulk modulus $K$ applies to all three (all resist volume change). Young's modulus and rigidity modulus are meaningful only for solids, since fluids cannot sustain tension or shear.

    Hint: Fluids: only bulk.

22. 22.Why do fluids have no modulus of rigidity?

    Fluids (liquids and gases) cannot sustain a shearing (tangential) stress at rest — they flow — so their shear strain is unlimited and $\eta\to 0$.

    Hint: They flow under shear.

23. 23.Define Poisson's ratio $\sigma$.

    Ratio of lateral strain to longitudinal strain: $\sigma=-\dfrac{\text{lateral strain}}{\text{longitudinal strain}}=-\dfrac{\Delta d/d}{\Delta L/L}$. It is dimensionless.

    Hint: Sideways vs lengthwise strain.

24. 24.What is the theoretical range of Poisson's ratio?

    Theoretically $-1<\sigma<0.5$; for most real materials $\sigma$ lies between $0.2$ and $0.4$.

    Hint: Practical: 0.2–0.4.

25. 25.What is lateral strain?

    The fractional change in a transverse dimension (e.g. diameter) when a wire is stretched: $\dfrac{\Delta d}{d}$. A stretched wire gets thinner.

    Hint: Sideways contraction on stretching.

26. 26.Relation between $Y$, $K$ and $\sigma$.

    $Y=3K(1-2\sigma)$.

    Hint: Connects Young's, bulk, Poisson.

27. 27.Relation between $Y$, $\eta$ and $\sigma$.

    $Y=2\eta(1+\sigma)$.

    Hint: Connects Young's, rigidity, Poisson.

28. 28.Express Poisson's ratio in terms of $\eta$ and $K$.

    $\sigma=\dfrac{3K-2\eta}{2(3K+\eta)}$.

    Hint: Eliminate Y from the two relations.

29. 29.Give the relation among $Y$, $K$ and $\eta$.

    $\dfrac{9}{Y}=\dfrac{1}{K}+\dfrac{3}{\eta}$, equivalently $Y=\dfrac{9K\eta}{3K+\eta}$.

    Hint: Reciprocal-type relation.

30. 30.For $\sigma=0.5$, what is special about the material?

    $Y=3K(1-2\sigma)=0$ when $\sigma=0.5$, so the material is perfectly incompressible (no volume change; $K\to\infty$ relatively).

    Hint: Upper limit of σ.

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