Question

12. How is constructed the deformation tensor?

Answer 1

The deformation gradient F is the fundamental measure of deformation in continuum mechanics. It is the second order tensor which maps line elements in the reference configuration into line elements (consisting of the same material particles) in the current configuration. Consider a line element dX emanating from position X in the reference configuration which becomes dx in the current configuration, Fig. 2.2.1. Then, using

dx= (X + dX) - (X)

=(Grad )dX

A capital G is used on “Grad” to emphasise that this is a gradient with respect to the material coordinates1, the material gradient,/X

The motion vector-function , is a function of the variable X, but it is customary to denote this simply by x, the value of at X, i.e. x=x(X,'t), so that

  

F = x/X =Grad x , FiJ = xi/XJ

with

dx = FdX, dxi = FiJ dXJ

Lower case indices are used in the index notation to denote quantities associated with the spatial basis {e_i}i e whereas upper case indices are used for quantities associated with the material basis {E_I}

Note that

dx = x/X * dX

is a differential quantity and this expression has some error associated with it; the error (due to terms of order (dX)² and higher, neglected from a Taylor series) tends to zero as the differential dX 0 . The deformation gradient (whose components are finite) thus characterises the deformation in the neighbourhood of a point X, mapping infinitesimal line elements dX emanating from X in the reference configuration to the infinitesimal line elements dx emanating from x in the current configuration.

Homogeneous Deformations A homogeneous deformation is one where the deformation gradient is uniform, i.e. independent of the coordinates, and the associated motion is termed affine. Every part of the material deforms as the whole does, and straight parallel lines in the reference configuration map to straight parallel lines in the current configuration, as in the above example. Most examples to be considered in what follows will be of homogeneous deformations; this keeps the algebra to a minimum, but homogeneous deformation analysis is very useful in itself since most of the basic experimental testing of materials, e.g. the uniaxial tensile test, involve homogeneous deformations.

Rigid Body Rotations and Translations

One can add a constant vector c to the motion, x = x + c , without changing the deformation, Grad( x + c)= Gradx . Thus the deformation gradient does not take into account rigid-body translations of bodies in space. If a body only translates as a rigid body in space, then F = I , and x = X + c (again, note that F does not tell us where in space a particle is, only how it has deformed locally). If there is no motion, then not only is F = I , but x = X .

If the body rotates as a rigid body (with no translation), then F = R , a rotation tensor . For example, for a rotation of about the X₂ axis,

F=

Note that different particles of the same material body can be translating only, rotating only, deforming only, or any combination of these.

The Inverse of the Deformation Gradient

The inverse deformation gradient F^-1 carries the spatial line element dx to the material line element dX. It is defined as

F-1 = x/X =Grad X , FiJ = XI/xJ

.........

dx = F-1dX, dxi = FiJ-1 dXj ...................action of F-1

FF = FF = I F_iM FMj^-1 = _ij

The Cauchy-Green Strain Tensors

The deformation gradient describes how a line element in the reference configuration maps into a line element in the current configuration. It has been seen that the deformation gradient gives information about deformation (change of shape) and rigid body rotation, but does not encompass information about possible rigid body translations. The deformation and rigid rotation will be separated shortly. To this end, consider the following strain tensors; these tensors give direct information about the deformation of the body. Specifically, the Left Cauchy-Green Strain and Right Cauchy-Green Strain tensors give a measure of how the lengths of line elements and angles between line elements (through the vector dot product) change between configurations.