Stress (physics)

In physics, stress is the internal distribution of forces within a body that balance and react the loads applied to it.

Stress in one dimension

The idea of stress originates in two simple, but important, observations of the loading (in tension) of a one-dimensional body, for example, a steel wire.

1. For small loads, the deformation (strain) of the wire is proportional to the applied load divided by the cross-sectional area of the wire.
2. Failure occurs when the load exceeds a critical value for the material (the tensile strength) multiplied by the cross-sectional area of the wire.

These observations suggest that the fundamental characteristic that affects the deformation and failure of materials is stress, force divided by the area over which it is applied.

For instance, a steel bolt of diameter 5 mm, has a cross-sectional area of 19.63mm². A load of 50 N induces a stress (force distributed across the cross-section) of 50/19.63=2.55 MPa. This can be thought of as each m² of bolt supporting 2.55 MN of the total load. In another bolt with half the diameter, and hence a quarter the cross-sectional area, carrying the same 50 N load, the stress will be quadrupled (10.2 MPa).

Ultimate tensile strength is a property of a material and, for any particular geometry, it allows the calculation of the load that would cause fracture. The compressive strength is a similar property for compressive loads. The yield strength is the value of stress causing plastic deformation. These values are determined experimentally using the measurement procedure known as the tensile test.

Cauchy's principle

Augustin Louis Cauchy ennunciated the principle that, within a body, the forces that an enclosed volume imposes on the remainder of the material must be in equilibrium with the forces upon it from the rest of the body.

This intuition provides a route to characterising and calculating complicated patterns of stress. To be exact, the stress at a point may be determined by taking the limit of the load being carried by a particular cross section, divided by that cross section, as the area of the cross section approaches zero. In general the stress may vary from point to point, but for simple cases, such as circular cylinders with pure axial loading, the stress is constant and equal to the cross-sectional area divided by the applied load.We can consider a small element of the body that has an area ΔA, over which a force ΔP acts. By making the element indefinitely small and taking the limit:

- where σ is the stress. Because the force is a vector, the stress has two components, one in the plane of the area, A, the shear stress, and one perpendicular, the normal stress.

Plane stress

Plane stress is a two-dimensional state of stress in a body. This is a good model when a flat thin body is loaded in the plane of the body. A small volume element in equilibrium experiences forces in balance.

By specifying some co-ordinates, the forces P and Q can be resolved normal and perpendicular to the faces of the volume element.

The stresses on the element are:

• Normal stresses:
  • σ_x = P_x/ΔA
  • σ_y = Q_y/ΔA
• Shear stresses:
  • τ_xy = P_y/ΔA
  • τ_yx = Q_x/ΔA

Principal stresses

It is always possible to transform the co-ordinates on the body into a set in which the shear stress vanishes. The remaining normal stresses are called the principal stresses.

Mohr's circle

A graphical method for analysing plane stress was proposed by O. Mohr. - to be completed

Stress in three dimensions

The considerations above can be generalised to three dimensions.

Stress tensor

Because the behaviour of a body does not depend on the co-ordinates used to measure it, stress can be described by a tensor. The stress tensor is symmetric and can always be resolved into the sum of two symmetric tensors:

• A mean or hydrostatic stress tensor, involving only pure tension and compression; and
• A shear stress tensor, involving only stress.

In the case of a fluid, Pascal's Law shows that the hydrostatic stress is the same in all directions, at least to a first approximation, so can be captured by the scalar quantity pressure.

Stress measurement

As with force, stress cannot be measured directly but is usually inferred from measurements of strain and knowledge of elastic properties of the material.

Units

The SI unit for stress is the pascal (symbol Pa); in US Customary units, stress is expressed in pounds per square inch (psi).

Residual stress

To be done

Bibliography

See also

Stress-energy tensor Stress concentration

Books

• Dieter, G.E. (1988) Mechanical Metallurgy, ISBN 0071004068