Stress Components

Infinitely many planes can pass through a point , each with its own normal and corresponding traction vector . The totality of these pairs defines the state of stress at . In the next section, we will show that the state of stress can be completely determined by knowing the traction vectors on just three mutually perpendicular planes. That is, if we know the components of the traction vector on three mutually planes, we can determine the traction vector on any arbitrary plane.

Let the unit normals of these three planes be the basis vectors , , and .[^fn2] The traction vector on the plane perpendicular to the -axis is . For simplicity, we denote as . Since each is a vector, it has three components in the basis:

We denote the -th component of the vector as . These nine scalar values are the components of the Cauchy stress tensor. They are typically arranged in a matrix:

If the normal of a plane is , the component is normal to the surface, while and are the shearing components.

In general, the diagonal components (, , ) are called normal stresses, while the off-diagonal components where (, , etc.) are called shear stresses.

Other Notations

Other common notations for the stress tensor matrix include:

​ and the engineering notation, which uses for normal stresses and for shear stresses:

Units of Stress

In the international system or SI:

• Force measured in newtons (N)

• Area measured in square meters (m)

• Because stress is force divided by area, stress is measured in N/m. This unit is called a pascal and denoted by Pa:

• Because one pascal is quite small, stress is often measured in

  • kPa = kilo (10) pascal,
  • MPa = mega (10) pascal
  • GPa = giga (10) pascal

In the U.S. customary system:

• Force is measured in pounds (lb)
• Area is measured in square inches (in)
• Stress is expressed in pounds per square inch (psi) or kilopounds per square inch (ksi)

Sign Convention

A face is defined as positive if its outward normal vector aligns with a positive coordinate axis, and negative if its outward normal aligns with a negative coordinate axis.

A stress component is considered positive if it acts in a positive coordinate direction on a positive face, or if it acts in a negative direction on a negative face. A key consequence of this convention is that tensile normal stresses are positive and compressive stresses are negative.