Deformation (physics)

In physics and continuum mechanics, deformation is the change in the shape or size of an object.It is quantified as the residual displacement of particles in a non-rigid body, from an initial configuration to a final configuration, excluding the body's average translation and rotation (its rigid transformation).A deformation can occur because of external loads,[2] intrinsic activity (e.g. muscle contraction), body forces (such as gravity or electromagnetic forces), or changes in temperature, moisture content, or chemical reactions, etc.In a continuous body, a deformation field results from a stress field due to applied forces or because of some changes in the conditions of the body.The relation between stress and strain (relative deformation) is expressed by constitutive equations, e.g., Hooke's law for linear elastic materials.In this case, the continuum completely recovers its original configuration.On the other hand, irreversible deformations may remain, and these exist even after stresses have been removed.One type of irreversible deformation is plastic deformation, which occurs in material bodies after stresses have attained a certain threshold value known as the elastic limit or yield stress, and are the result of slip, or dislocation mechanisms at the atomic level.Deformation is the change in the metric properties of a continuous body, meaning that a curve drawn in the initial body placement changes its length when displaced to a curve in the final placement.If none of the curves changes length, it is said that a rigid body displacement occurred.On the other hand, the components xi of the position vector x of a particle in the deformed configuration, taken with respect to the spatial coordinate system of reference, are called the spatial coordinates There are two methods for analysing the deformation of a continuum.where x is the position of a point in the deformed configuration, X is the position in a reference configuration, t is a time-like parameter, F is the linear transformer and c is the translation.In matrix form, where the components are with respect to an orthonormal basis,A rigid body motion is a special affine deformation that does not involve any shear, extension or compression.The transformation matrix F is proper orthogonal in order to allow rotations but no reflections.A change in the configuration of a continuum body results in a displacement.A rigid-body displacement consists of a simultaneous translation and rotation of the body without changing its shape or size.Deformation implies the change in shape and/or size of the body from an initial or undeformed configuration κ0(B) to a current or deformed configuration κt(B) (Figure 1).The vector joining the positions of a particle P in the undeformed configuration and deformed configuration is called the displacement vector u(X,t) = uiei in the Lagrangian description, or U(x,t) = UJEJ in the Eulerian description.A displacement field is a vector field of all displacement vectors for all particles in the body, which relates the deformed configuration with the undeformed configuration.It is convenient to do the analysis of deformation or motion of a continuum body in terms of the displacement field.In general, the displacement field is expressed in terms of the material coordinates aswhere αJi are the direction cosines between the material and spatial coordinate systems with unit vectors EJ and ei, respectively.It is common to superimpose the coordinate systems for the undeformed and deformed configurations, which results in b = 0, and the direction cosines become Kronecker deltas:The partial differentiation of the displacement vector with respect to the material coordinates yields the material displacement gradient tensor ∇Xu.Similarly, the partial differentiation of the displacement vector with respect to the spatial coordinates yields the spatial displacement gradient tensor ∇xU.Homogeneous (or affine) deformations are useful in elucidating the behavior of materials.Some homogeneous deformations of interest are Linear or longitudinal deformations of long objects, such as beams and fibers, are called elongation or shortening; derived quantities are the relative elongation and the stretch ratio.Volume deformation is a uniform scaling due to isotropic compression; the relative volume deformation is called volumetric strain.From the polar decomposition theorem, the deformation gradient, up to a change of coordinates, can be decomposed into a stretch and a rotation.