Stress (physics)

In physics, stress is the internal distribution of forces within a body that balance and react to the loads applied to it. Stress is a tensor quantity with nine terms, but can be described fully by six terms due to symmetry. Simplifying assumptions are often used to represent stress as a vector for engineering calculations.

Stress tensor

Main article: Stress tensor.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Because the behavior of a body does not depend on the coordinates 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:

Related Topics:
Behavior - Coordinate - Tensor

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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

Related Topics:
Fluid - Pascal's law - Scalar - Pressure

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Thus, in the case of a solid, the hydrostatic (or isostatic) pressure p is defined as one third of the trace of the tensor, i.e., the mean of the diagonal terms.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:p = rac{mathrm{tr}(T)}{3} = rac{sigma_{11} + sigma_{22} + sigma_{33}}{3}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Generalized notation

In the generalized stress tensor notation, the tensor components are written σij, where i and j are in {1;2;3}.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The first step is to number the sides of the cube.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

When the lines are parallel to a vector base (ec{e_1},ec{e_2},ec{e_3}), then:

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

• the sides perpendicular to are called j and -j; and
• from the center of the cube, points toward the j side, while the -j side is at the opposite.

σij is then the component along the i axis that applies on the j side of the cube. (Or in books in the English language, σij is the stress on the i face acting in the j direction -- the transpose of the subscript notation herein. But transposing the subscript notation produces the same stress tensor, since a symmetric matrix is equal to its transpose.)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

This generalized notation allows an easy writing of equations of the continuum mechanics, such as the generalized Hooke's law:

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:sigma_{ij} = sum_{kl} C_{ijkl} cdot arepsilon_{kl}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The correspondence with the former notation is thus:

~ ~ ~ ~ ~ ~ ~ ~ ~ ~