Elasticity, Stress & Strain
Internal forces per area, and how much the body gives.
Lesson at a glanceShow the key points
- Elasticity: the tendency of a deformed body to regain its shape. Perfectly elastic bodies recover fully; perfectly plastic ones don't recover at all.
- Microscopically, displaced molecules feel restoring forces from their neighbours, like a particle held by springs.
- Stress = internal force per unit area: normal (longitudinal: tensile or compressive) Fₙ/ΔS, shearing Fₜ/ΔS (14.1, 14.2), volume stress = pressure (14.3).
- Longitudinal strain ΔL/L (14.4); shearing strain = displacement of a layer ÷ its distance from the fixed layer (x/h); volume strain ΔV/V.
Our “rigid bodies” were an idealisation. Pull on any real rod and it stretches a little; twist it and it twists. How much, and how it springs back, depends on the material.
Elasticity
When forces deform a body, internal forces appear that try to restore its natural shape. The property of resisting deformation and restoring shape is elasticity.
- A body that regains its shape completely when the forces are removed is perfectly elastic.
- One that stays deformed and doesn’t recover at all is perfectly plastic (inelastic).
- Most bodies are partially elastic: they recover part of the way.
Stress
Let a body be acted on by forces that add to zero, so it stays put but deforms. Imagine a small cross-section ΔS inside it. The parts on the two sides push or pull on each other with forces F and −F. Split F into a component Fₙ normal to ΔS and a component Fₜ along it:
Longitudinal stress is tensile when the two parts pull on each other (a stretched wire; Fₙ is the tension) and compressive when they push (a rod squeezed at both ends). Unless stated otherwise, the cross-section is taken perpendicular to the length.
Volume stress
A body immersed in a fluid at pressure P feels a force PΔS on every surface element, always perpendicular to the surface. This force per unit area is the volume stress, and it simply equals the pressure (14.3).
A 10 kg mass hangs from a copper wire of radius 1.0 mm. Find the tensile stress in the wire.
Strain
Each kind of stress has a matching measure of deformation, a strain (a pure number):
In shear, the four tangential forces on a block add to zero and so do their torques, so the block stays in equilibrium while it deforms.
A rubber pad 5.0 cm thick is fixed at its base. A sideways force on its top face moves that face 2.0 mm. What is the shearing strain?
- 1Elasticity: deformed bodies develop restoring forces; bodies range from perfectly elastic to perfectly plastic.
- 2Stress = internal force per unit area: normal (tensile/compressive), shearing, or volume (= pressure).
- 3Strain = fractional deformation: ΔL/L, x/h, ΔV/V.
Check yourself
Keep exploring
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