Recover the force on a cut from Cauchy stress

Picture cutting a loaded body into two pieces and replacing the removed piece by the force it exerted across the cut. At a smooth point define that force per current area as the traction $\mathbf t(\mathbf n)$, where $\mathbf n$ points outward from the retained piece. Under the ordinary Cauchy continuum assumptions, local force balance implies a linear dependence on the unit normal. The resulting tensor is a local field; it can vary through the body even when a total applied force is constant.

Our convention is $t_i=\sigma_{ij}n_j$, with repeated Cartesian indices summed. Thus column j is the traction on the face with outward normal along positive axis j. The entry $\sigma_{xy}$ is the x-directed force per area on a y-normal face. Some sources reverse the order when naming shear components, so inspect the traction equation rather than guessing from a subscript. Every stress component has units of pascals, or newtons per square metre.

Project the traction onto the cut normal to obtain normal stress. Subtract that normal vector part to get tangential or shear traction. A diagonal stress matrix in one coordinate system can still produce shear traction on an oblique plane. Shear is therefore not synonymous with an off-diagonal entry unless the plane is aligned with the chosen axes. The physically testable statement is about a specified surface normal and the force transmitted across it.

With tensile normal stress taken positive, uniform hydrostatic pressure has $\boldsymbol\sigma=-p\mathbf I$ for $p>0$. Its traction is inward on every outward-facing surface and has no shear component on any plane. A uniaxially stretched bar behaves differently: only its axial stress is nonzero in the ideal interior, so oblique planes generally carry both normal and shear traction. Principal stress directions are the orientations for which the traction is parallel to the normal and the shear part vanishes.

For a classical continuum without independent body couples or couple stresses, angular momentum balance makes the Cauchy stress symmetric. Opposite shear tractions then create no unbalanced local torque in the limit of a shrinking element. This symmetry reduces the number of independent entries from nine to six in three dimensions. It follows from a balance law and the chosen continuum structure, not from a definition that every possible stress-like object must satisfy.

Stress is a tensor rather than a collection of unrelated loads. Rotating Cartesian axes changes its components by a two-sided rotation, while the traction on the same physical plane remains the same physical vector. Stress also differs from total force: the average axial stress in a uniform bar is force divided by area, but near a hole or a load introduction point the distribution is nonuniform. A correct global force measurement alone cannot reveal the largest local stress or establish material safety.