Ebenenstress

Ebenenstress, known in English as plane stress, denotes a two‑dimensional state of stress in which the stresses perpendicular to a thin dimension are neglected. Specifically, for a plate or film of small thickness, the out-of-plane normal stress sigma_z and the shear stresses tau_xz and tau_yz are assumed to be zero (sigma_z = 0, tau_xz = tau_yz = 0). This simplification reduces a three‑dimensional elasticity problem to an effectively two‑dimensional one.

For an isotropic linear elastic material under plane stress, the in‑plane stresses relate to the in‑plane strains by:

sigma_x = (E/(1 − nu^2)) (epsilon_x + nu epsilon_y)

sigma_y = (E/(1 − nu^2)) (epsilon_y + nu epsilon_x)

tau_xy = G gamma_xy, with G = E/(2(1 + nu)) and gamma_xy = 2 epsilon_xy.

Equivalently, tau_xy = (E/(1 + nu)) epsilon_xy. Here E is Young’s modulus, nu is Poisson’s ratio, and G is the shear modulus. These relations reflect how in‑plane stresses respond to in‑plane strains while out‑of‑plane components are neglected.

Applications commonly include thin metal sheets, composite skins, printed circuit boards, and other structures where thickness is small relative to in‑plane dimensions. Plane stress is often used in finite element analysis to obtain 2D approximations of 3D problems and to simplify design calculations.

Limitations arise when the thickness is not small, when through‑thickness gradients or bending induce significant sigma_z or tau_xz/tau_yz, or under plastic deformation and large strains. In such cases, plane stress may not be an adequate model, and plane strain or full 3D analysis may be required. Plane stress is contrasted with plane strain, where epsilon_z ≈ 0 and sigma_z adjusts to maintain zero axial strain.