# Stiffness and hardness , strength and toughness

#### Cosmossos

Hello
can someone please tell me the difference between stiffness and hardness and between strength and toughness?

thanks

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#### tiny-tim

Homework Helper
Hello Cosmossos!
Hello
can someone please tell me the difference between stiffness and hardness
Sorry, don't know.
… and between strength and toughness?
Breaking strength (of a material) is force per area (stress) just before failure (in N/m2)

Toughness (of a material) is energy per volume just before failure (in J/m3).

But isn't energy = force times displacement, so energy per volume = force times displacement per volume = force times area? 1 J/m3 = 1 N/m2 ?

Yeees, but energy is the integral of force times displacement, so the total energy (per volume) absorbed by the material before failure depends on the shape of the force-displacement (per volume) curve (more usually called the strain-stress curve)

(strain = ∫ displacement per thickness = ∫ displacement times area per volume,
and stress = force per area,
so strain-stress = ∫ force times displacement per volume = energy per volume)​

For good examples of stress-strain curves, see http://www.etomica.org/app/modules/sites/MaterialFracture/Images/SSPicture2.jpg" [Broken]
on the page http://www.etomica.org/app/modules/sites/MaterialFracture/Background1.html" [Broken] …

the linear part is the elastic region, where energy per volume is proportional to force per area, the non-linear (curved) part is the plastic region (the ductile region, if we're talking about tension), and it ends at failure …

the total energy absorbed by the material (the area under the graph) depends on the shape of that non-linear part.

(it curves down at the end because of the difference between apparent stress and actual stress … the actual stress-strain curve keeps going up … see http://upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Stress_v_strain_A36_2.svg/300px-Stress_v_strain_A36_2.svg.png" [Broken])

Brittle materials are strong but not tough … they fail almost immediately after the end of the linear part.

Yield strength (of a material) is force per area at the top of the linear part.

If we bothered to define yield toughness (of a material), it would be energy per volume at the top of the linear part, but that would simply be proportional to yield strength, so we don't bother.

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