These pages deal with basic materials science and physical properties of materials.

About Stress and Strain

P Pallav, PhD

Strain

Deformation occurs when materials are loaded with a mechanical stress or force. In the literature, the words strain, deflection, and distortion are used as synonyms. There are two kinds of strain:

• Elastic deformation is defined as the (part of the) deformation, which is reversible. This is the part of the deformation that springs back when the load is removed.
• Plastic deformation is the irreversible part of the deformation. This is the deformation, which remains after the load has been removed.

Although the word strain may be used for absolute strain, ∆ (delta) in the figure, in which case it is expressed in meters (or millimeters), it generally refers to relative strain, ε (epsilon),

ε = ∆/L₀ (equation 1)

Because it is calculated as a length (∆) divided by length (L₀) or meters over meters, it is dimensionless, i.e. it is a fraction, which may optionally be expressed in percents, parts per million (ppm), etc. 

In the case of a 10-meter long string, which is elongated by 20 millimeters (=0.02m), the strain equals 0.002, 0.2%, or 2000ppm.

Contraction Ratio

When a specimen is stretched, the diameter decreases. The amount of this transverse contraction is proportional not only to the elongation, ε, but also to ν (nu), the contraction ratio of the material. Transverse contraction ratio, Poisson’s ratio, or constant are synonyms of this material property.

The contraction ratio can only have a value between 0 and 0.5, i.e. 0<ν<0.5.

It must be greater than 0, because all materials, which are stretched, contract perpendicular to this direction.

The contraction ratio must be smaller than 0.5, because a value of 0.5 means that the volume wouldn’t change, because the transverse contraction compensates the elongation.

If no transverse contraction would occur, i.e. when ν would be zero, stretching the specimen by 2% would increase the volume by 2%. With ν=0.5, the diameter decreases 1% (0.5x2%) and because this contraction also occurs square to the plane of drawing, this would exactly compensate the 2%-increase. Such in-expandable or incompressible materials do not exist.

Most materials have contraction ratios between 0.2 and 0.35. In many cases ν can safely be rounded to 0.25.

A steel string with ν=0.25 and a diameter of 2 millimeter, which is stretched (ε) by 0.2%, becomes 0.05% (εxν) or 1 micrometer thinner. The 3.14mm² area of the cross-section decreases 0.1% (-2xεxν), so there is a net increase in volume of 0.1% (0.2-0.1).

Perhaps more than the elastic module, the contraction ratio is related to the internal coherence of the material. Discontinuities and especially porosities (e.g. cast iron) lead to lesser values of n.

Stress

Force, F, or stress, σ (sigma), gives rise to deformation or vice versa. Forces are expressed in Newtons (N). A mass of 1-kilogram (~2.2lbs.) weighs about 10N, and a pound (~0.45kg) about 4.5 Newtons here on earth. Stress is the force (N) per unit area (m²) and is similar to pressure. Its unit, Newtons per square meter has been named the Pascal (1Pa = 1N/m²), but MegaPascals (1MPa=10⁶Pa) are more convenient for mechanical stresses. It may be of practical use to remember that 1MPa=1N/mm².

In a simple tensile experiment like in the figure above, the stress (σ) in the sample is equal to the force divided by the area of the cross-section (A).

σ = F / A (equation 2)

A string with a diameter (D) of 2 millimeter has a cross-section (A) of 3.14mm² (A=pxR² with R=½D=1mm and p=3.14) and a load (F) of 1256N would produce a stress of (1256/3.14=) 400MPa.

Principal Stresses

The principal stresses are parameters, which are used to assess the load conditions in a particular point in a structure.
Normally six stress components are needed to completely define a load condition in a point in an xyz-coordinate system; the three stresses, σ_x, σ_y, and σ_z, and the three shear stresses, t_xy, t_xz, and t_yz.

The figure below illustrates the difference between stress and shear stress, the two elementary kinds of loading. Because this sketch is 2D, it can only show two stresses (σ_x, and σ_y) and one shear stress (t_xy).

Stress versus Shear stress

As shown in the figure for the xy plane, shear stresses may be converted to stresses by changing the orientation of the coordinate system. It is always possible to align a 3D coordinate system in such a way that all three shear stresses disappear (i.e. become zero) and that only three perpendicular tensile or compressive stresses completely describe the load condition at any point.

These are called the principal stresses, σ₁, σ₂, and σ₃. The indices 1-3 refer to the three mutually perpendicular axes of the rotated coordinate system and are ranked in order of magnitude. Therefore σ₁ is the greatest of the three, which is also called the main principal stress. Failure of a structure is to be expected at the point where the greatest, i.e. the maximum main principal stress occurs.

In the simple case of a string, which is loaded with a tension of 400MPa, the main principal stress (σ₁) is equal to 400MPa and the other principal stresses (σ₂, σ₃) are zero.