The deformation of a thin straight rod into a closed loop. The length of the rod remains almost unchanged during the deformation, which indicates that the strain is small. In this particular case of bending, displacements associated with rigid translations and rotations of material elements in the rod are much greater than displacements associated with straining.
Deformations which cease to exist after the stress field is removed are termed aselastic deformation. 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 isplastic deformation, which occurs in material bodies after stresses have attained a certain threshold value known as theelastic limit oryield stress, and are the result ofslip, ordislocation mechanisms at the atomic level. Another type of irreversible deformation isviscous deformation, which is the irreversible part ofviscoelastic deformation.In the case of elastic deformations, the response function linking strain to the deforming stress is thecompliance tensor of the material.
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 arigid body displacement occurred.
It is convenient to identify a reference configuration or initial geometric state of the continuum body which all subsequent configurations are referenced from. The reference configuration need not be one the body actually will ever occupy. Often, the configuration att = 0 is considered the reference configuration,κ0(B). The configuration at the current timet is thecurrent configuration.
For deformation analysis, the reference configuration is identified asundeformed configuration, and the current configuration asdeformed configuration. Additionally, time is not considered when analyzing deformation, thus the sequence of configurations between the undeformed and deformed configurations are of no interest.
The componentsXi of the position vectorX of a particle in the reference configuration, taken with respect to the reference coordinate system, are called thematerial or reference coordinates. On the other hand, the componentsxi of the position vectorx of a particle in the deformed configuration, taken with respect to the spatial coordinate system of reference, are called thespatial coordinates
There is continuity during deformation of a continuum body in the sense that:
The material points forming a closed curve at any instant will always form a closed curve at any subsequent time.
The material points forming a closed surface at any instant will always form a closed surface at any subsequent time and the matter within the closed surface will always remain within.
Anaffine deformation is a deformation that can be completely described by anaffine transformation. Such a transformation is composed of alinear transformation (such as rotation, shear, extension and compression) and a rigid body translation. Affine deformations are also calledhomogeneous deformations.[3]
Therefore, an affine deformation has the formwherex 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 andc is the translation. In matrix form, where the components are with respect to an orthonormal basis,
The above deformation becomesnon-affine orinhomogeneous ifF =F(X,t) orc =c(X,t).
A rigid body motion is a special affine deformation that does not involve any shear, extension or compression. The transformation matrixF isproper orthogonal in order to allow rotations but noreflections.
A rigid body motion can be described bywhereIn matrix form,
A change in the configuration of a continuum body results in adisplacement. The displacement of a body has two components: a rigid-body displacement and a deformation. 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 configurationκ0(B) to a current or deformed configurationκt(B) (Figure 1).
If after a displacement of the continuum there is a relative displacement between particles, a deformation has occurred. On the other hand, if after displacement of the continuum the relative displacement between particles in the current configuration is zero, then there is no deformation and a rigid-body displacement is said to have occurred.
The vector joining the positions of a particleP in the undeformed configuration and deformed configuration is called thedisplacement vectoru(X,t) =uiei in the Lagrangian description, orU(x,t) =UJEJ in the Eulerian description.
Adisplacement 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 asor in terms of the spatial coordinates aswhereαJi are the direction cosines between the material and spatial coordinate systems with unit vectorsEJ andei, respectively. Thusand the relationship betweenui andUJ is then given by
Knowing thatthen
It is common to superimpose the coordinate systems for the undeformed and deformed configurations, which results inb = 0, and the direction cosines becomeKronecker deltas:
Thus, we haveor in terms of the spatial coordinates as
The partial differentiation of the displacement vector with respect to the material coordinates yields thematerial displacement gradient tensor∇Xu. Thus we have:orwhereF is thedeformation gradient tensor.
Similarly, the partial differentiation of the displacement vector with respect to the spatial coordinates yields thespatial displacement gradient tensor∇xU. Thus we have,or
Linear or longitudinal deformations of long objects, such as beams and fibers, are calledelongation orshortening; derived quantities are therelative elongation and thestretch ratio.
Plane deformations are also of interest, particularly in the experimental context.
Volume deformation is a uniform scaling due to isotropiccompression; the relative volume deformation is calledvolumetric strain.
A plane deformation, also calledplane strain, is one where the deformation is restricted to one of the planes in the reference configuration. If the deformation is restricted to the plane described by the basis vectorse1,e2, thedeformation gradient has the formIn matrix form,From thepolar decomposition theorem, the deformation gradient, up to a change of coordinates, can be decomposed into a stretch and a rotation. Since all the deformation is in a plane, we can write[3]whereθ is the angle of rotation andλ1,λ2 are theprincipal stretches.
Asimple shear deformation is defined as an isochoric plane deformation in which there is a set of line elements with a given reference orientation that do not change length and orientation during the deformation.[3]
Ife1 is the fixed reference orientation in which line elements do not deform during the deformation thenλ1 = 1 andF·e1 =e1.Therefore,Since the deformation is isochoric,DefineThen, the deformation gradient in simple shear can be expressed asNow,Sincewe can also write the deformation gradient as
