Deformationsfeld

Deformationsfeld, or deformation field, is a vector field that assigns to every material point in a body the displacement it undergoes relative to a reference configuration. It is a central concept in continuum mechanics, materials science, and geophysics. If x denotes a point in the reference configuration and u(x) its displacement, the current position is x' = x + u(x).

From the displacement field one defines the deformation gradient F = ∂x'/∂x = I + ∇u, which encodes local stretches and rotations of material elements. For small deformations, the linear strain tensor ε = (∇u + ∇u^T)/2 describes the symmetric part of the displacement gradient. For large deformations, the Green–Lagrange strain tensor E = 1/2 (F^T F − I) is used to measure nonlinear changes in shape and size. Material behavior is then described by constitutive laws relating stress and the chosen strain measure.

In practice, deformation fields are measured or estimated by various methods. In geoscience and geodesy, techniques such as Interferometric Synthetic Aperture Radar (InSAR) and Global Positioning System (GPS) data yield deformation fields of the Earth's surface. In experimental mechanics and image analysis, digital image correlation (DIC) and optical flow estimate two- or three-dimensional deformation fields from sequences of images.

Applications of the deformation field span structural health monitoring, metal forming, earthquake and crustal deformation studies, and the validation of numerical simulations in solid mechanics. The mathematical properties of a deformation field—continuity, differentiability, and compatibility with boundary conditions—are essential for correct physical interpretation and modeling.

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