Summary of equilibrium and elasticity

Conditions for equilibrium: For a rigid body to be in equilibrium, two conditions must be satisfied. First, the vector sum of forces must be zero. Second, the sum of torques about any point must be zero. The torque due to the weight of a body can be found by assuming the entire weight is concentrated at the center of gravity, which is at the same point as the center of mass if g S has the same value at all points.

Stress, strain, and Hooke’s law: Hooke’s law states that in elastic deformations, stress (force per unit area) is proportional to strain (fractional deformation). The proportionality constant is called the elastic modulus.

Tensile and compressive stress: Tensile stress is tensile
force per unit area, F/A. Tensile strain is fractional
change in length, ?l/l₀. The elastic modulus for tension
is called Young’s modulus Y. Compressive stress and
strain are defined in the same way. 

Bulk stress: Pressure in a fluid is force per unit area.
Bulk stress is pressure change, ?p, and bulk strain is
fractional volume change, ?V/V0. The elastic modulus
for compression is called the bulk modulus, B.
Compressibility, k, is the reciprocal of bulk modulus:
k = 1/B.

Shear stress: Shear stress is force per unit area,
FŒ/A, for a force applied tangent to a surface. Shear
strain is the displacement x of one side divided by
the transverse dimension h. The elastic modulus
for shear is called the shear modulus, S.

The limits of Hooke’s law: The proportional limit is the maximum stress for which stress and strain
are proportional. Beyond the proportional limit, Hooke’s law is not valid. The elastic limit is the
stress beyond which irreversible deformation occurs. The breaking stress, or ultimate strength, is
the stress at which the material breaks.

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