Advanced elasticity/Time derivatives and rates

Time derivatives and rate quantities edit

Material time derivatives edit

Material time derivatives are needed for many updated Lagrangian formulations of finite element analysis.

Recall that the motion can be expressed as


If we keep   fixed, then the velocity is given by


This is the material time derivative expressed in terms of  .

The spatial version of the velocity is


We will use the symbol   for velocity from now on by slightly abusing the notation.

We usually think of quantities such as velocity and acceleration as spatial quantities which are functions of   (rather than material quantities which are functions of  ).

Given the spatial velocity  , if we want to find the acceleration we will have to consider the fact that  , i.e., the position also changes with time. We do this by using the chain rule. Thus


Such a derivative is called the material time derivative expressed in terms of  . The second term in the expression is called the convective derivative..

Velocity gradient edit

Let the velocity be expressed in spatial form, i.e.,  . The spatial velocity gradient tensor is given by


The velocity gradient   is a second order tensor which can expressed as


The velocity gradient is a measure of the relative velocity of two points in the current configuration.

Time derivative of the deformation gradient edit

Recall that the deformation gradient is given by


The time derivative of   (keeping   fixed) is


Using the chain rule


Form this we get the important relation


Time derivative of strain edit

Let   and   be two infinitesimal material line segments in a body. Then




Taking the derivative with respect to   gives us


The material strain rate tensor is defined as






The spatial rate of deformation tensor or stretching tensor is defined as


In fact, we can show that   is the symmetric part of the velocity gradient, i.e.,


For rigid body motions we get  .

Lie derivatives edit

Most of the operations above can be interpreted as push-forward and pull-back operations. Also, time derivatives of these tensors can be interpreted as Lie derivatives.

Recall that the push-forward of the strain tensor from the material configuration to the spatial configuration is given by


The pull-back of the spatial strain tensor to the material configuration is given by


Therefore, the rate of deformation tensor is a push-forward of the material strain rate tensor, i.e.,


Similarly, the material strain rate tensor is a pull-back of the rate of deformation tensor to the material configuration, i.e.,






Therefore the rate of deformation tensor can be obtained by first pulling back   to the reference configuration, taking a material time derivative in that configuration, and then pushing forward the result to the current configuration.

Such an operation is called a Lie derivative. In general, the Lie derivative of a spatial tensor   is defined as


Spin tensor edit

The velocity gradient tensor can be additively decomposed into a symmetric part and a skew part:


We have seen that   is the rate of deformation tensor. The quantity   is called the spin tensor.

Note that   is symmetric while   is skew symmetric, i.e.,


So see why   is called a "spin", recall that










So we have






The second term above is invariant for rigid body motions and zero for an uniaxial stretch. Hence, we are left with just a rotation term. This is why the quantity   is called a spin.

The spin tensor is a skew-symmetric tensor and has an associated axial vector   (also called the angular velocity vector) whose components are given by




The spin tensor and its associated axial vector appear in a number of modern numerical algorithms.

Rate of change of volume edit

Recall that


Therefore, taking the material time derivative of   (keeping   fixed), we have


At this stage we invoke the following result from tensor calculus:

If   is an invertible tensor which depends on   then


In the case where   we have






Alternatively, we can also write


These relations are of immense use in numerical algorithms - particularly those which involved incompressible behavior, i.e., when  .