Jugami Physics

High School · Physics · Unit 20

Materials: Stress, Strain and Elasticity

Why steel cables hold up bridges, why glass snaps and rubber stretches, and how engineers predict exactly how much a material will give.

Every bridge, building, bone and bike frame is a material under load. Pull on a steel wire and it stretches a tiny amount, then springs back. Pull harder and it stretches for good. Pull harder still and it breaks. Physics lets you predict each step.

On this page you'll meet the three kinds of load, the ideas of stress and strain, and Young's modulus, the number that says how stiff a material is. Then you'll read a stress-strain curve, compare brittle and ductile materials and see how bridges, bones and spider silk use them. After that you can try the stretch calculator, read the history and take a quiz.

Start with forcesJump to the calculator
A suspension bridge: two towers hold up a curved main cable, and thin hanger cables drop from it to carry the road deck.

On this page

Key words first

Tension

A load that pulls a material apart and makes it longer.

Compression

A load that pushes a material together and makes it shorter.

Shear

A load that pushes one face sideways relative to the opposite face, changing the shape.

Stress (σ)

Force per area: σ = F ÷ A, in pascals (Pa). 1 MPa = 10⁶ Pa and 1 GPa = 10⁹ Pa.

Strain (ε)

Fractional change in length: ε = ΔL ÷ L. It has no units and is often given as a percent.

Young's modulus (E)

Stiffness: stress ÷ strain in the elastic region. A big E means a stiff material.

Elastic

Goes back to its original shape when the load is removed.

Plastic

Stays permanently stretched or bent after the load is removed.

Idea 1: Tension, compression and shear

A material can be loaded in three basic ways. In tension forces pull it apart, as in a rope holding a climber or a cable holding up a bridge. In compression forces squeeze it, as in a column holding up a roof or your leg bones holding up you. In shear forces push neighboring layers sideways past each other, as in a bolt joining two plates or the blades of scissors.

Most real parts carry a mix. A beam bending under a load is in compression along its top and tension along its bottom. Materials can behave very differently under each: concrete is strong in compression but weak in tension, which is why it's reinforced with steel bars.

Three ways to load a material. Tension: forces pull a block apart and it gets longer. Compression: forces push it together and it gets shorter. Shear: forces push the top and bottom opposite ways and the block leans over. Cables are in tension, columns in compression and bolts in shear.tensionpulled apart:gets longercompressionpushed together:gets shortershearpushed sideways:changes shapecable, rope, hanger: tensioncolumn, tower, leg bone: compressionrivet, bolt, scissors: shear
Figure 1. The three basic loads and where you meet them.

Idea 2: Stress and strain

A thick rope can hold more than a thin one, so engineers compare materials using the force per area, the stress:

σ = F ÷ A

A long wire stretches more than a short one under the same load, so they also use the fractional stretch, the strain:

ε = ΔL ÷ L

Stress is measured in pascals, the same unit as pressure. Strain has no units: a 2.000 m wire that stretches 1.0 mm has a strain of 0.000 50, or 0.050%. Solids stretch very little before something gives; strains in steel structures are usually well under 0.1%.

A rod fixed to a wall, shown before and after a force F pulls its free end. Its original length L grows by a small extra length delta L. The force spreads over the rod's cross-section area A. Stress is F divided by A; strain is delta L divided by L.Fbefore: length LLΔLarea Astress σ = F ÷ Astrain ε = ΔL ÷ L
Figure 2. A force F on a rod of area A stretches its length L by ΔL.

Idea 3: Young's modulus

For small loads, stress and strain are proportional. That's Hooke's law, the same rule as for a spring, written for a material instead of an object. The constant of proportionality is Young's modulus:

E = σ ÷ ε = FL ÷ (A ΔL)

Rearranged, it tells you how much a wire or rod stretches: ΔL = FL ÷ (AE). Because strain has no units, E is in pascals, usually gigapascals. Steel's modulus is about 200 GPa; rubber's is thousands of times smaller, which is why it stretches so easily. The same formula works for compression, with ΔL as the amount the rod shortens.

Young's modulus of nine materials on a log scale where each grid line is ten times the one before, in gigapascals: rubber 0.01 to 0.1, nylon 66 2.93, beech wood 9.5 to 11.9, bone 14, high-strength concrete 30, aluminium 68, copper 110, steel 200 and diamond 1050 to 1210.0.010.11101001000rubber0.01–0.1nylon 662.93wood (beech)9.5–11.9bone14concrete30aluminium68copper110steel200diamond1050–1210Young's modulus E (GPa, log scale)
Figure 3. Young's modulus on a log scale, where each grid line is ten times the one before. Bars show a range. Values from Wikipedia's "Young's modulus."

Checked values

MaterialYoung's modulus (GPa)Yield strength (MPa)Tensile strength (MPa)
Structural steel (ASTM A36)200250400–550
Copper (99.9%)11070220
Aluminium (alloy 6061-T6 for strengths)68241300
Float glass47.7–83.6——
High-strength concrete30—2–5 (concrete)
Bone (limb)14104–121130
Wood (American beech)9.5–11.9——
Nylon 662.93——
Rubber (small strain)0.01–0.1——
Spider silk——1,000
Diamond (synthetic)1050–1210——

Moduli from Wikipedia's "Young's modulus"; strengths from Wikipedia's "Ultimate tensile strength." A dash means not listed there. Real samples vary with grade, treatment and, for wood and bone, direction.

Idea 4: The stress-strain curve

Pull a metal bar in a testing machine and plot stress against strain and you get a curve with landmarks:

  1. Elastic region and elastic limit. The curve starts as a steep straight line, where Hooke's law holds and the bar springs back fully. Its slope is Young's modulus. The elastic limit is the end of that region.
  2. Yield point. Just past the elastic limit the metal starts to yield: it stretches permanently. For structural steel this is about 250 MPa.
  3. Plastic region and ultimate strength. The bar keeps stretching with more load, up to the highest stress it can take, its ultimate tensile strength, 400–550 MPa for structural steel.
  4. Necking and fracture. After that a narrow neck forms, the load it can carry falls and the bar fractures.

Engineers keep working stresses well below the yield point, using a safety factor, so structures always stay in the elastic region.

A stress-strain curve for a ductile metal such as steel. It starts as a steep straight line, the elastic region, up to point 1, the elastic limit. Just above is point 2, the yield point, where the metal begins to stretch permanently. The curve then rises more slowly through the plastic region to its peak, point 3, the ultimate strength. After that the bar necks down and the curve falls to point 4, fracture.stressstraingreen: elastic region1 elastic limit2 yield point3 ultimate strength4 fractureplastic: stretches for goodneckingstraight part: stress ∝ strain (Hooke's law)
Figure 4. A stress-strain curve for a ductile metal, not to scale. The elastic region is shaded green.

Idea 5: Brittle and ductile

A ductile material, such as copper or mild steel, has a long plastic region. It bends and stretches a lot before breaking, which gives warning and soaks up energy. That's why copper can be drawn into wire and why car bodies are made to crumple.

A brittle material, such as glass, cast iron or concrete, has almost no plastic region. It stays elastic right up to the moment it cracks, then fails suddenly. Brittle materials are often fine in compression but dangerous in tension, where a tiny crack can spread fast.

The same material can change. Some steels turn brittle in the cold: during World War II, US Liberty ships had nearly 1,500 cases of serious brittle fracture.

Stress-strain curves for a brittle and a ductile material. The brittle one, like glass, rises in a straight line and snaps at a small strain. The ductile one, like copper, rises, then bends over and keeps stretching a long way before it breaks, absorbing much more energy.stressstrainbrittle (glass, cast iron)snaps with almost no stretchductile (copper, mild steel)stretches a lot, then breaksthe area under each curve ∝ energy absorbed
Figure 5. Brittle materials snap at small strain. Ductile ones stretch a long way first.

Idea 6: Strength is not stiffness

Stiffness (Young's modulus) tells you how much a material stretches under a load. Strength tells you how much stress it can take before it yields or breaks. They're different: glass is about as stiff as aluminium but breaks far more easily, and spider silk is much less stiff than steel yet has a higher tensile strength.

The breaking force of a rod or cable is its strength times its area, F = σ × A. Double a cable's diameter and you get four times the area, so four times the strength.

Tensile strength, the stress that breaks a material in tension, in megapascals: concrete 2 to 5, limb bone 130, copper 220, aluminium alloy 6061 300, structural steel 400 to 550 and spider silk about 1,000.02505007501,000concrete2–5bone (limb)130copper220aluminium 6061300structural steel400–550spider silk1,000tensile strength (MPa)light blue: range for different grades
Figure 6. Tensile strength, the stress that breaks a material when pulled. Values from Wikipedia's "Ultimate tensile strength."

Idea 7: Bridges, buildings, bones and silk

Bridges. In a suspension bridge the main cables and hangers are in tension and the towers are in compression. Steel wire is ideal in tension: each main cable of the Golden Gate Bridge is made of 27,572 steel wires.

Buildings. Concrete is cheap and strong when squeezed, but its tensile strength is only about 2–5 MPa. Embedding steel bars, called rebar, lets the steel take the tension, which is why most modern buildings are reinforced concrete.

Bones. Limb bone has a modulus of about 14 GPa and a tensile strength of about 130 MPa, a light, tough mix of stiff mineral and flexible collagen. Leg bones mostly work in compression, but a fall can bend or twist them and load one side in tension.

Spider silk. Dragline silk has a tensile strength around 1,000 MPa, higher than structural steel's 400–550 MPa, while being far less dense. Researchers are working to spin artificial silk to match it.

A suspension bridge. The main cable hangs between two towers and holds up the deck through vertical hangers, so the cable and hangers are in tension. The cable's pull presses down on the towers, so the towers are in compression.tower: compressionmain cable: tensiondecksteel wire cables carry the pull;concrete and steel towers carry the push
Figure 7. In a suspension bridge the cables pull and the towers push.

Worked examples

Try each one first, then open the solution. Use g = 9.8 N/kg and the values in the table above.

1. Stress in a cable

A cable with a cross-section of 5.0 × 10⁻⁵ m² holds 2 000 N. What's the stress?

Show solution
σ = 2 000 ÷ (5.0 × 10⁻⁵)
= 4.0 × 10⁷ Pa = 40 MPa

That's well below steel's 250 MPa yield point.

2. Strain

A 2.000 m wire stretches by 1.0 mm. What's the strain?

Show solution
ε = 0.001 0 ÷ 2.000
= 5.0 × 10⁻⁴ (0.050%)

Remember to put both lengths in the same unit.

3. How far does it stretch?

A steel wire 2.0 m long and 1.0 mm across holds 100 N. How much does it stretch?

Show solution
A = π × (0.000 50)² = 7.85 × 10⁻⁷ m²
ΔL = (100 × 2.0) ÷ (7.85 × 10⁻⁷ × 200 × 10⁹)
≈ 1.27 mm

Use the radius, half the diameter, for the area.

4. Find the modulus

A 1.0 m rod with area 1.0 × 10⁻⁴ m² stretches 1.5 mm under 10 000 N. What's E, and what might it be made of?

Show solution
E = (10 000 × 1.0) ÷ (1.0 × 10⁻⁴ × 0.001 5)
≈ 6.67 × 10¹⁰ Pa = 66.7 GPa

Close to aluminium's 68 GPa.

5. A thigh bone

A thigh bone's solid cross-section is about 3.0 cm². What's the stress when it carries 700 N of body weight?

Show solution
A = 3.0 × 10⁻⁴ m²
σ = 700 ÷ (3.0 × 10⁻⁴) ≈ 2.33 MPa

About 1.4% of bone's compressive strength of about 170 MPa, so the bone is nowhere near its limit. Running and jumping raise it several times.

6. Before it yields

A structural steel rod is 10 mm across. What's the biggest load it can carry before it yields?

Show solution
A = π × (0.005 0)² = 7.85 × 10⁻⁵ m²
F = 250 × 10⁶ × 7.85 × 10⁻⁵
≈ 19.6 kN

That's the weight of about 2 000 kg. With a safety factor, you'd load it far less.

Try it: Stretch calculator

Wire and rod stretch calculator

How the science developed

People built with stone, wood and rope for thousands of years by trial and error. The science of how materials stretch and break began in the 1600s.

  1. 1638 Galileo's Two New Sciences treats the strength of materials as one of its two new sciences.
  2. 1678 Robert Hooke publishes his law as the Latin Ut tensio, sic vis: "as the extension, so the force."
  3. 1727 Leonhard Euler develops the idea behind the modulus of elasticity; Giordano Riccati measures it in experiments in 1782.
  4. 1807 Thomas Young describes the modulus that now carries his name.
  5. 1856 Henry Bessemer patents a process for making steel cheaply, opening the way to steel bridges and skyscrapers.
  6. 1883 The Brooklyn Bridge opens, the first steel-wire suspension bridge.
  7. 1937 The Golden Gate Bridge opens, hung from two cables of 27,572 wires each.
  8. 1940s Brittle fractures in welded Liberty ships push engineers to study why steel can turn brittle and crack.
  9. Today Engineers design with composites such as carbon fibre and model structures on computers, and labs are learning to spin artificial spider silk.

Test yourself

The Materials Check

Talk about it

  1. Why do you think concrete floors and bridges have steel bars inside them?
  2. A glass and a plastic cup fall off a table. Use the words brittle and ductile to explain what happens.
  3. Why might a doctor put a cast on a broken arm rather than let you keep using it?
  4. If spider silk is so strong, why don't we build bridges out of it?

Sources

  1. OpenStax, College Physics 2e, Section 5.3 "Elasticity: Stress and Strain."
  2. OpenStax, University Physics Volume 1, Section 12.3 "Stress, Strain, and Elastic Modulus" and Section 12.4 "Elasticity and Plasticity."
  3. Wikipedia, "Young's modulus," "Ultimate tensile strength," "Stress–strain curve," "Ductility," "Bone" and "Spider silk."
  4. Wikipedia, "Robert Hooke," "Two New Sciences," "Bessemer process," "Brooklyn Bridge," "Golden Gate Bridge" and "Liberty ship."

Credits: all diagrams on this page, including the load diagrams, the stretched rod, the modulus and strength charts, the stress-strain curves, the bridge and the share image, are original SVG made for Jugami.