Time derivatives and rate quantities
Material time derivatives
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
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
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 the relative velocity of two points in the
Time derivative of the deformation gradient
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
Let and be two infinitesimal material line segments in a
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
In fact, we can show that is the symmetric part of the velocity
For rigid body motions we get .
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
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
The spin tensor and its associated axial vector appear in a number of modern
Rate of change of volume
Therefore, taking the material time derivative of (keeping fixed),
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 .