# Rate of Deformation and Spin Tensors

The most obvious first step is to take the time derivative of the deformation gradient, , as follows:

For reasons that won’t become clear until the chapter on rate-form constitutive expressions, we would like the “velocity gradient” to be expressed with respect to rather than . Since is a function of , we can use the chain rule to arrive at the following important expression:

(1)

We can see from eq. 1 that
or
, where is the “velocity gradient.”

For our purposes, consider that if we were to have analytical functions of time, , for each term in that describes how and are changing with time, for example, then we could find by noting that and . This is quite academic (and potentially quite difficult) compared to the way that would be found in FEA, so such an example will not be given. However, is a very important quantity as we will see in the chapter on rate-form constitutive relationships.

The velocity gradient, , is a very important quantity and can be decomposed as follows:

(2)

note: Similar to eq. 2, in linear infinitesimal elasticity, we can split the displacement gradient into a strain tensor and rotation tensor that are symmetric and anti-symmetric, respectively.

note: Any tensor can be similarly split into its symmetric and anti-symmetric parts.

“Rate of Deformation Tensor” = symmetric
“Spin Tensor” = anti-symmetric or “skew”

is a pure rigid body rotation. A complete proof of this can be found in [Asaro]. It can also be easily shown that is, in fact, skew. This proof can be found in Appendix A.2.

Consider the time rate of change of length of a particular element:

So,

(3)

While [Holzapfel] shows that “ is not pure rate of strain and is not a pure rate of rotation,” we can see that the rate of change of length only depends on , not . Additionally, if then we must have rigid body motions only.

note: We could have split up into a sum, but we wouldn’t have arrived at anything useful. The way that eq. 3 was derived only works due to the product rule of derivatives. Proving that represents “spin” is more difficult and won’t be shown here, although the derivation can be found in many other texts (e.x. [Asaro][Bonet][Holzapfel]).

Similar to the shear strain, , we can again consider how the angle between two vectors, , and , changes under deformation (see Fig below).

(4)

note: (Recall also: )
reduces to for infinitesimal deformation (ignoring rigid body rotations), since .
In general, though, . Similarly, in general, . We’ll also look in detail at the physical meaning (or lack thereof) of later on when we get to the chapter on “Rate-Form Constitutive Expressions”

• R. J. Asaro and V. A. Lubarda, Mechanics of Solids and Materials, Cambridge, UK: Cambridge University Press, 2006.
[Bibtex]
@BOOK{Asaro,
Author         = {Robert J. Asaro and Vlado A. Lubarda},
Edition        = {},
Publisher      = {Cambridge University Press},
Title          = {Mechanics of Solids and Materials},
Year           = {2006}
}
• G. Holzapfel, Nonlinear Solid Mechanics, John Wiley & Sons Ltd., England, 2000.
[Bibtex]
@book{Holzapfel,
title={Nonlinear Solid Mechanics},
author={Holzapfel, GA},
year={2000},
publisher={John Wiley \& Sons Ltd., England}
}
• J. Bonet and R. Wood, Nonlinear Continuum Mechanics for Finite Element Analysis. 1997, Cambridge University Press, Cambridge.
[Bibtex]
@book{Bonet,
title={Nonlinear {C}ontinuum {M}echanics for {F}inite {E}lement {A}nalysis. 1997},
author={Bonet, J and Wood, RD},
publisher={Cambridge University Press, Cambridge}
}