# Elasticity

## Summary

- Generalized Hooke's law
- stress
- A stress is a force or combination of forces distributed throughout the whole of an object that acts to deform it.
- Stresses take the general form of force divided by area (
*F*/*A*). - The SI unit of stress is the pascal or newton per meter sqared [Pa = N/m
^{2}]

- strain
- A strain is any deformation of an object measured geometrically as a fraction of some original value.
- Strains take the general form of a change in one geometric quantity divided by the original value of that quantity or a similar quantity with the same unit (∆ℓ/ℓ
_{0}, ∆*V*/*V*_{0}, etc.). - Strains are always dimensionless or unitless ratios [m/m, m
^{3}/m^{3}, etc.]

- modulus (plural, moduli)
- Stress is directly proportional to strain.
- An elastic modulus is the ratio of some stress to the strain caused by that stress.
- The SI unit of all elastic moduli is the pascal or newton per meter squared [Pa = N/m
^{2}], but gigapascals [1 GPa = 10^{9}Pa] are more commonly used. - Elastic moduli are properties of materials, not the objects made from those materials.

- stress
- Tension and compression
- stress
- Tensile stress is the normal force per area (σ =
*F*/*A*) that causes an object to increase in length. - Compressive stress is the normal force per area (σ =
*F*/*A*) that causes an object to decrease in length.

- Tensile stress is the normal force per area (σ =
- strain
- Tensile strain is the fractional increase in length of an object (ε = ∆ℓ/ℓ
_{0}) due to a tensile stress. - Compressive strain is the fractional decrease in length of an object (ε = ∆ℓ/ℓ
_{0}) due to a compressive stress. - modulus
- Young's modulus or elastic modulus is the ratio of tensile stress to tensile strain or compressive stress to compressive strain.
- The symbol for Young's modulus is
*E*(for*élasticité*) or*Y*(for Young).*F*= *E*∆ℓ *A*ℓ _{0}σ = *E*ε

- Poisson's ratio
- Axial strain in one sense is usually accompanied by transverse strain in the opposite sense.
- Tensile stress makes objects longer and thinner.
- Compressive stress makes objects shorter and fatter.

- The negative ratio of transverse strain (∆
*y*/*y*_{0}or ∆*z*/*z*_{0}) to axial strain (∆*x*/*x*_{0}) is called Poisson's ratio.- The symbol for Possion's ratio is
*ν*(nu).ν = − ∆ *y*/*y*_{0}= − ∆ *z*/*z*_{0}∆ *x*/*x*_{0}∆ *x*/*x*_{0}

- The symbol for Possion's ratio is

- Axial strain in one sense is usually accompanied by transverse strain in the opposite sense.

- stress
- Shear
- stress
- Shear stress is the tangential force per area (τ =
*F*/*A*) that causes one face of an object to become displaced parallel to the opposite face.- Shear stress changes rectangles into parallelograms.

- Shear stress is the tangential force per area (τ =
- strain
- Shear strain is the fractional tangential displacement relative to the normal distance between opposite parallel faces (γ = ∆
*x*/*y*) caused by a shear stress.- Shear strain is the tangent of the shear angle.

- Shear strain is the fractional tangential displacement relative to the normal distance between opposite parallel faces (γ = ∆
- modulus
- The shear modulus or rigidity modulus is the ratio of shear stress to shear strain.
- The symbol for shear modulus is
*G*(for*glissement*) or*S*(for shear)*F*= *G*∆ *x**A**y*τ = *G*γ

- The symbol for shear modulus is

- The shear modulus or rigidity modulus is the ratio of shear stress to shear strain.

- stress
- Bulk
- stress
- Pressure is the compressive stress (
*P*=*F*/A) applied uniformly to all surfaces of an object.- Uniform compression or decompression changes the volume of objects but not their shape.

- Pressure is the compressive stress (
- strain
- Volume strain is the fractional change in volume of an object (θ = ∆
*V*/*V*_{0}) due to a change in pressure.

- Volume strain is the fractional change in volume of an object (θ = ∆
- modulus
- The bulk modulus or compression modulus is the ratio of the increase in pressure to the relative decrease in volume.
- The symbol for bulk modulus is
*K*(for*kompression*) or*B*(for bulk).*F*= *K*∆ *V**A**V*_{0}*P*=*Κ*θ

- The symbol for bulk modulus is
- The reciprocal of bulk modulus is called compressibility.
- The symbol for compressibility is β (beta) or κ (kappa).
β = 1 *K* - The SI unit of compressibility is the inverse pascal [Pa
^{−1}].

- The symbol for compressibility is β (beta) or κ (kappa).

- The bulk modulus or compression modulus is the ratio of the increase in pressure to the relative decrease in volume.

- stress