What is the equation for the Traction Vector?

What is the equation for the Traction Vector?

• A. Stress Tensor Times Unit Normal Vector
• B. Displacement Vector Times Stress Tensor
• C. Stress Tensor Times Velocity Vector
• D. Pressure Times Area Vector

Answer: (A)

Traction vector is the force per unit area that acts on a surface, and its value depends on how the surface is oriented. The way to get it is to multiply the Cauchy stress tensor by the unit normal to the surface: t = σ · n. Here σ is the 3x3 stress tensor, and n is the outward unit normal to the surface. In components, this is t_i = σ_{ij} n_j, which shows how the stress state in the material projects onto that particular direction. This formulation accounts for both normal and shear components of the surface traction. The unit normal is essential because it fixes the surface orientation; the total force on a small surface element is dF = t dA = σ n dA, with dA being the element area. If the stress were purely hydrostatic, σ = -p I, then t = -p n, which reduces to pressure acting normal to the surface. Other expressions like displacement times stress, velocity times stress, or pressure times an area vector do not generalize traction on an arbitrary surface in the same way and miss the directional and shear information captured by t = σ · n.

Traction vector is the force per unit area that acts on a surface, and its value depends on how the surface is oriented. The way to get it is to multiply the Cauchy stress tensor by the unit normal to the surface: t = σ · n. Here σ is the 3x3 stress tensor, and n is the outward unit normal to the surface. In components, this is t_i = σ_{ij} n_j, which shows how the stress state in the material projects onto that particular direction.

This formulation accounts for both normal and shear components of the surface traction. The unit normal is essential because it fixes the surface orientation; the total force on a small surface element is dF = t dA = σ n dA, with dA being the element area.

If the stress were purely hydrostatic, σ = -p I, then t = -p n, which reduces to pressure acting normal to the surface. Other expressions like displacement times stress, velocity times stress, or pressure times an area vector do not generalize traction on an arbitrary surface in the same way and miss the directional and shear information captured by t = σ · n.