Defining Stress
This section summarizes the definition of different stress measures, stress invariants, and other important definitions.
Three different stress measures are used in COMSOL Multiphysics:
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Cauchy stress σ defined as force/deformed area in fixed spatial directions not following the body. This is a symmetric tensor.
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First Piola-Kirchhoff stress P. The forces in the spatial directions are related to the area in the original (material) frame. This is an unsymmetric two-point tensor.
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Second Piola-Kirchhoff stress S. Both force and area are represented in the material configuration. For small strains the values are the same as Cauchy stress tensor but the directions are rotating with the body. This is a symmetric tensor.
The stresses relate to each other as
In a geometrically linear analysis, the distinction between the stress measures disappear, and they all converge to the same values.
In the documentation, the symbol σ is used to denote not only Cauchy stress, but stress in general. The symbols P and S are used whenever it is necessary to make a distinction. In geometrically nonlinear analysis, the stress should in general be interpreted as second Piola-Kirchhoff stress.
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The internal variables for the Cauchy stresses are named solid.sx, solid.sxy, and so on.
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The internal variables for the first Piola-Kirchhoff stresses are named solid.PxX, solid.PxY, and so on.
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The internal variables for the second Piola-Kirchhoff stresses are named solid.SX, solid.SXY, and so on.
Sign Conventions
A positive normal stress in COMSOL Multiphysics acts in tension. This is the most widely used definition in general physics and engineering.
Within the field of geomechanics it is however common to let compressive stresses be positive, since compression is almost always dominant in that field of science. When working with the material models intended for soils and rocks, you must be aware of that the “positive in tension” convention is used also there in order to maintain consistency within the software.
Specifically, the ordering of Principal Stresses is such that (including signs). In geotechnical applications the dominant principal stresses will usually be compressive, so the third principal stress will the be the one which you may consider as “largest”.
The convention used in Ref. 1 refers to the hydrostatic pressure (trace of the stress Cauchy tensor) with a positive sign. The use of the first invariant of Cauchy stress tensor I1(σ) is preferred in this document where there is a risk of confusion. The pressure in COMSOL Multiphysics is always defined as positive under compression, or equivalently, it has the opposite sign of the Cauchy stress tensor’s trace.