Glossary term
Stress Tensor
The mathematical object that fully describes the stress state at a point inside a three-dimensional body.
Definition
conceptThe stress tensor is a second-order symmetric tensor that completely characterises the stress state at a point inside a deformable body, specifying the normal and shear stresses acting on every possible internal surface through that point.
A single scalar value is insufficient to describe stress at a point in a three-dimensional body because stress depends on the orientation of the surface on which it acts. The stress tensor resolves this by collecting all stress components — three normal stresses and six shear stresses, reduced to six independent values by symmetry — into a single mathematical object. From the stress tensor, engineers can derive principal stresses, apply multiaxial failure criteria, and compute stress in any direction using coordinate transformations.
At a point inside a three-dimensional body, stress cannot be described by a single number. A uniaxial force produces different internal effects depending on the orientation of the cross-section being examined — the same point experiences different normal and shear stresses depending on how the cutting plane is oriented. The stress tensor resolves this ambiguity by specifying the complete stress state at a point, from which the stress on any surface can be computed.
Components of the stress tensor
With respect to a Cartesian coordinate system (x, y, z), the Cauchy stress tensor \boldsymbol{\sigma} is written as a 3 \times 3 matrix:
The diagonal components \sigma_{xx}, \sigma_{yy}, \sigma_{zz} are normal stresses acting perpendicular to the faces of an infinitesimal element aligned with the coordinate axes. The off-diagonal components \tau_{xy}, \tau_{xz}, \tau_{yz} (and their counterparts) are shear stresses acting parallel to those faces. By the angular momentum balance of a material element, the tensor is symmetric: \tau_{xy} = \tau_{yx}, \tau_{xz} = \tau_{zx}, \tau_{yz} = \tau_{zy}. This symmetry reduces nine components to six independent values.
Traction vector and Cauchy’s formula
The stress tensor connects the stress state at a point to the traction vector \mathbf{t} — the force per unit area — acting on any surface with outward unit normal \mathbf{n}. Cauchy’s formula states:
This linear relationship is the defining property of the stress tensor: it maps a surface orientation to the corresponding traction. Given the full tensor, the traction on any plane through the point is immediately computable.
Principal stresses
There exist three mutually perpendicular orientations — the principal directions — for which the shear stresses on the corresponding faces are zero. The normal stresses on these faces are the principal stresses \sigma_1 \geq \sigma_2 \geq \sigma_3. They are the eigenvalues of the stress tensor, found by solving:
Principal stresses are invariant with respect to coordinate rotation — they characterise the intrinsic stress state at the point, independent of the chosen reference frame. They are the natural inputs to failure criteria such as the von Mises criterion and the Tresca criterion, and to graphical representations such as Mohr’s circle.
Stress invariants
Three scalar quantities derived from the stress tensor are invariant under coordinate transformations. The first invariant is the trace:
It is proportional to the mean hydrostatic stress p = I_1 / 3, which drives volumetric deformation. The second and third invariants (I_2, I_3) appear in the characteristic equation for the principal stresses and in yield criteria formulated in terms of the deviatoric stress tensor.
Engineering use
The stress tensor is the fundamental output of any stress analysis — analytical, semi-analytical, or numerical. Finite element software computes the six independent stress components at every integration point in a model and derives from them the quantities needed for design checks: principal stresses, von Mises equivalent stress, maximum shear stress, and hydrostatic pressure. Understanding the tensor structure is essential for interpreting stress results correctly, especially in components with complex geometry or multiaxial loading where uniaxial intuition fails.
Common mistakes
A common mistake is treating the largest plotted finite element stress component as the design stress without checking coordinate system, averaging, singularities, element quality, and the relevant failure criterion. Another is comparing principal stress, von Mises stress, and shear stress as if they were interchangeable. A strong stress-tensor review states coordinate frame, sign convention, stress measure, extraction location, averaging method, load combination, material model, and the criterion used for acceptance.