Elasticity

Definition of Modulus of Elusticity:

It is often difficult to precisely define yielding due to the wide variety of stress–strain curves exhibited by real materials. In addition, there are several possible ways to define yielding:^[1]
True elastic limit: The lowest stress at which dislocations move. This definition is rarely used, since dislocations move at very low stresses, and detecting such movement is very difficult.
Proportionality limit:Up to this amount of stress, stress is proportional to strain (Hooke's law), so the stress-strain graph is a straight line, and the gradient will be equal to the elastic modulus of the material.

Elastic limit (yield strength):Beyond the elastic limit, permanent deformation will occur. The elastic limit is therefore the lowest stress at which permanent deformation can be measured. This requires a manual load-unload procedure, and the accuracy is critically dependent on the equipment used and operator skill. For elastomers, such as rubber, the elastic limit is much larger than the proportionality limit. Also, precise strain measurements have shown that plastic strain begins at low stresses.^[2][3]

Yield point:The point in the stress-strain curve at which the curve levels off and plastic deformation begins to occur.^[4]

Offset yield point (proof stress):When a yield point is not easily defined based on the shape of the stress-strain curve an offset yield point is arbitrarily defined. The value for this is commonly set at 0.1 or 0.2% strain.^[5] The offset value is given as a subscript, e.g., R_p0.2=310 MPa.^[6] High strength steel and aluminum alloys do not exhibit a yield point, so this offset yield point is used on these materials.^[5]

Upper and lower yield points:Some metals, such as mild steel, reach an upper yield point before dropping rapidly to a lower yield point. The material response is linear up until the upper yield point, but the lower yield point is used in structural engineering as a conservative value. If a metal is only stressed to the upper yield point, and beyond, Lüders bands can develop.^[7]