tractievector

Tractievector, or traction vector in English, is the force per unit area that acts on a specific plane inside a continuous medium. At a point, consider a plane with unit normal n; the traction vector t(n) is the limit of the interfacial force per unit area as the plane area shrinks to zero. In continuum mechanics, the traction on that plane is given by t(n) = σ n, where σ is the Cauchy stress tensor. The traction thus depends on the plane’s orientation and position, and the tensor σ fully determines all possible tractions at that point. In Dutch-language texts, tractievector is the standard term; in English literature, traction vector is used.

Mathematically, the traction on a plane with normal n is t(n) = σ n. The Cauchy stress tensor σ is typically symmetric, and for a given point it encodes normal and shear stresses acting on all possible planes through that point. The normal component of the traction is t_n = t · n = n · σ · n, a scalar representing the load along the plane’s normal. The remaining part, t_t = t − t_n n, is the tangential or shear traction.

Applications and interpretation: Traction vectors are central to boundary conditions in elasticity and fluid mechanics. For a body occupying a volume V with boundary S, the traction on the boundary is t = σ n. The total force due to internal stresses on the boundary is the surface integral ∮_S t dS, and, by the divergence theorem, it equals ∫_V div σ dV (together with body forces as applicable). In engineering, tractions are used to assess loads, design boundaries, and analyze failure criteria, yielding, and material response under applied surfaces.