Stress and Strain Tensor Basics: Definition, Equations & Moduli

[Greg Bernhardt]

Definition/Summary

Stress is force per area, and is a tensor.

It is measured in pascals ($Pa$), with dimensions of mass per length per time squared ($ML^{-1}T^{-2}$).

By comparison, load is force per length, and strain is a dimensionless ratio, stressed length per original length.

Equations



Extended explanation

Stress and strain tensors:

The trace and traceless parts of the stress tensor (the Cauchy stress tensor) are the pressure tensor (a multiple of the unit tensor) and the shear tensor, respectively.

By comparison, the trace and traceless parts of the strain tensor are the volumetric strain tensor (a multiple of the unit tensor) and the strain deviator tensor, respectively.

Hooke's law:

The stress and strain tensors are second-order tensors, and are linearly related by a fourth-order Hooke tensor:

$$\sigma_{ij}\ =\ \sum_{kl} h_{ijkl}\,\varepsilon_{kl}$$

Modulus:

For isotropic material, the Hooke tensor may be replaced by two scalars (ordinary numbers), the bulk modulus and shear modulus.

A modulus is a ratio of stress to strain. It has the same dimensions and units as stress.

Stiffness is a property of a particular body. It is modulus times cross-section area per length. Young's modulus ($E$) is a form of stiffness.

Bulk modulus ($K$) of an isotropic material is the ratio of pressure to volumetric strain.

Shear modulus ($G$ or $\mu$) of an isotropic material is the ratio of the shear tensor to the strain deviator tensor.

Moment of area:

A moment of area measures a particular body's resistance to stress, relative to a particular axis. It depends only on shape, not on density.

Tangent modulus and secant modulus:

For the straight portion of the stress-strain graph (up to the proportional limit of the material), tangent modulus and secant modulus are the same.

At a general point on the graph, tangent modulus is the slope of the tangent, but secant modulus is the slope of the line joining the point to the origin.

In other words, tangent modulus is dstress/dstrain (the marginal stress/strain, or the local rate of stress per strain), but secant modulus is the total stress/strain.

See http://www.instron.co.uk/wa/resourcecenter/glossaryterm.aspx?ID=99 for a fuller explanation, and a diagram.

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!

 

[APhysicist]

Thanks for this great explanation about stress and strain tensors, their modulus, and their moments of area! This is very helpful for anyone trying to understand the basics of these concepts.