6. Rates of Deformation

The development of the last chapter fixed our attention on an instant \(t \in (0,T)\), and proposed some measurements of material deformation in terms of the configuration mapping \(\varphi_t\). We now allow time to vary and consider two questions:

• How are velocities and accelerations quantified in both the spatial and reference frames?

• How are time derivatives of deformation measures properly considered in a large deformation framework?

The former topic is obviously crucial in the formulation of dynamics problems, while the latter is necessary, for example, in rate-dependent materials where quantities such as strain rate must be quantified.

6.1. Material and Spatial Velocity and Acceleration

One obtains the material velocity \(\mathbf{V}\) and the material acceleration \(\mathbf{A}\) by fixing attention on a particular material particle (i.e., fixing the reference coordinate \(\mathbf{X}\)), and then considering successive (partial) time derivatives of the motion \(\varphi(\mathbf{X},t)\). This can be written mathematically as

(6.1)\[\mathbf{V}(\mathbf{X},t) = \frac{\partial}{\partial t} \big( \varphi(\mathbf{X},t) \big)\]

and

(6.2)\[\mathbf{A}(\mathbf{X},t) = \frac{\partial}{\partial t} \big( \mathbf{V}(\mathbf{X},t) \big) = \frac{\partial^2}{\partial t^2} \big( \varphi(\mathbf{X},t) \big) .\]

Note in (6.1) and (6.2) that \(\mathbf{V}\) and \(\mathbf{A}\) take \(\mathbf{X}\) as their first argument, hence their designation as material quantities. A Lagrangian description of motion, in which reference coordinates are the independent variables, would most naturally use these measures of velocity and acceleration.

An Eulerian description, on the other hand, generally requires measures written in terms of spatial points \(\mathbf{x}\) without requiring explicit knowledge of material points \(\mathbf{X}\). The spatial velocity \(\mathbf{v}\) and the spatial acceleration \(\mathbf{a}\) are obtained from (6.1) and (6.2) through a change in variables:

(6.3)\[\mathbf{v}(\mathbf{x},t) = \mathbf{V} \left(\varphi^{-1}_t (\mathbf{x}),t \right) = \mathbf{V}_t \cdot \varphi^{-1}_t (\mathbf{x})\]

and

(6.4)\[\mathbf{a}(\mathbf{x},t) = \mathbf{A} \left(\varphi^{-1}_t (\mathbf{x}),t \right) = \mathbf{A}_t \cdot \varphi^{-1}_t (\mathbf{x}) .\]

The expression given in (6.4) for the spatial acceleration may be unfamiliar to those versed in fluid mechanics who may be more accustomed to thinking of acceleration as the total time derivative of the spatial velocity \(\mathbf{v}\). We reconcile these different viewpoints here through the introduction of the equivalent concept of the material time derivative, defined, in general, as the time derivative of any object, spatial or material, taken so that the identity of the material particle is held fixed. Applying this concept to the spatial velocity gives

(6.5)\[\begin{split}\begin{aligned} \mathbf{a}(\mathbf{x},t) & = \left. \dot{\mathbf{v}}(\mathbf{x},t) \right|_{\mathbf{x}=\varphi(\mathbf{X},t)} \\ & = \left. \frac{d}{dt} \right|_{\mathbf{X} \text{ fixed}} \mathbf{v} \left( \varphi (\mathbf{X},t),t \right) \\ & = \frac{\partial \mathbf{v}}{\partial \mathbf{x}} (\mathbf{x},t) \cdot \frac{\partial \varphi}{\partial t} \left( \varphi^{-1}_t (\mathbf{x}),t \right) + \frac{\partial \mathbf{v}}{\partial t} \left( \varphi^{-1}_t (\mathbf{x}),t \right) \\ & = \frac{\partial \mathbf{v}}{\partial t} + \nabla \mathbf{v} \cdot \mathbf{v} . \end{aligned}\end{split}\]

This may be recognized as the so-called total time derivative of the spatial velocity \(\mathbf{v}\). Exercising the concept of a material time derivative a little further, we can see from (6.1) that the material velocity is the material time derivative of the motion, so that

(6.6)\[\mathbf{V} =\dot{\varphi} .\]

Comparing (6.2) and (6.5), we conclude that \(\mathbf{A}\) and \(\mathbf{a}\) are, in fact, the same physical entity expressed in different coordinates. The former is most naturally written in terms of \(\mathbf{V}\), while the latter is conveniently expressed in terms of \(\mathbf{v}\).

(6.5) uses the superposed dot notation for the time derivative of \(\mathbf{v}\). Such superposed dots will always imply a material time derivative in this document, whether applied to material or spatial quantities. Furthermore, the gradient \(\nabla \mathbf{v}\) is taken with respect to spatial coordinates and is called the spatial velocity gradient. It is used often enough to warrant the special symbol \(\mathbf{L}\):

(6.7)\[\mathbf{L} := \nabla \mathbf{v} .\]

6.2. Rate of Deformation Tensors

From the spatial velocity gradient \(\mathbf{L}\) defined in (6.7), we define two spatial tensors \(\mathbf{D}\) and \(\mathbf{W}\), known respectively as the spatial rate of deformation tensor and the spatial spin tensor:

(6.8)\[\mathbf{D} :=\nabla_s \mathbf{v} = \frac{1}{2} [\mathbf{L} + \mathbf{L}^T],\]

and

(6.9)\[\mathbf{W} :=\nabla_a \mathbf{v} = \frac{1}{2} [\mathbf{L} - \mathbf{L}^T].\]

It is clear that \(\mathbf{D}\) is merely the symmetric part of the velocity gradient, while \(\mathbf{W}\) is the antisymmetric, or skew, portion.

The quantities \(\mathbf{D}\) and \(\mathbf{W}\) are called spatial measures of deformation. \(\mathbf{D}\) is effectively a measure of strain rate suitable for large deformations, while \(\mathbf{W}\) provides a local measure of the rate of rotation of the material. In fact, in small deformations it is readily verified that (6.8) amounts to nothing more than the time derivative of the infinitesimal strain tensor defined in (2.5). It is of interest at this point to discuss whether appropriate material counterparts of these objects exist. Toward this end, we calculate the material time derivative of the deformation gradient \(\mathbf{F}\). If \(\mathbf{F}\) is an analytic function, the order of partial differentiation can be reversed:

(6.10)\[\dot{\mathbf{F}} = \frac{\partial}{\partial t} \left[ \frac{\partial}{\partial \mathbf{X}} \varphi (\mathbf{X},t) \right] = \frac{\partial}{\partial \mathbf{X}} \left[ \frac{\partial}{\partial t} \varphi (\mathbf{X},t) \right] = \frac{\partial \mathbf{V}}{\partial \mathbf{X}} .\]

From (6.10), we conclude that the material time derivative \(\dot{\mathbf{F}}\) is nothing more than the material velocity gradient. Manipulating this quantity further gives

(6.11)\[\frac{\partial \mathbf{V}}{\partial \mathbf{X}} = \frac{\partial}{\partial \mathbf{X}} \left(\mathbf{v}\circ \varphi _t \right) = \nabla \mathbf{v} \left( \varphi_t(\mathbf{X}) \right) \frac{\partial}{\partial \mathbf{X}} \left( \varphi_t(\mathbf{X}) \right) = \mathbf{L} \left( \varphi_t(\mathbf{X}) \right) \mathbf{F} (\mathbf{X}) .\]

Examination of (6.10) and (6.11) reveals that

(6.12)\[\mathbf{L} = \left( \dot{\mathbf{F}} \cdot \varphi^{-1}_t \right) \mathbf{F}^{-1} .\]

Recalling the definition for the right Cauchy-Green strain tensor \(\mathbf{C}\) in (5.14) Section 5, we compute its material time derivative via

(6.13)\[\dot{\mathbf{C}} = \frac{\partial}{\partial t} [\mathbf{F}^T \mathbf{F} ] = \dot{\mathbf{F}}^T \mathbf{F} + \mathbf{F}^T \dot{\mathbf{F}} = (\mathbf{L}\mathbf{F})^T \mathbf{F} + \mathbf{F}^T (\mathbf{L}\mathbf{F}) = \mathbf{F}^T(\mathbf{L}+\mathbf{L}^T)\mathbf{F} .\]

which, in view of (6.8), leads us to conclude

(6.14)\[\dot{\mathbf{C}}(\mathbf{X},t) = 2 \mathbf{F}^T (\mathbf{X},t) \mathbf{D}(\varphi(\mathbf{X}),t) \mathbf{F}(\mathbf{X},t) .\]

(6.14) reveals why \(\frac{1}{2}\dot{\mathbf{C}}\) is sometimes called the material rate of deformation tensor. Noting that \(\mathbf{F}\) is the Jacobian of the transformation \(\varphi_t\), readers with a background in differential geometry will recognize \(\frac{1}{2}\dot{\mathbf{C}}\) as the pull-back of the spatial tensor field \(\mathbf{D}\) defined on \(\varphi_t(\Omega)\). Conversely, \(\mathbf{D}\) is the push-forward of the material tensor field \(\frac{1}{2}\dot{\mathbf{C}}\) defined on \(\Omega\). The concepts of pull-back and push-forward are outside the scope of this document, but the physical principle they embody in the current context is perhaps useful. Loosely speaking, the push forward (or pull-back) of a tensor with respect to a given transformation produces a tensor in the new frame of reference that we, as observers, would observe as identical to the original tensor if we were embedded in the material during the transformation. Thus, the same physical principle is represented by both \(\frac{1}{2}\dot{\mathbf{C}}\) and \(\mathbf{D}\), but they are very different objects mathematically since the transformation that interrelates them is the deformation itself. Recalling the definition of Green’s strain \(\mathbf{E}\) given in (5.19), we can easily see that

(6.15)\[\dot{\mathbf{E}} = \frac{1}{2} \dot{\mathbf{C}} = \mathbf{F}^T\mathbf{D} \mathbf{F} .\]

This further substantiates the interpretation of \(\mathbf{D}\) as a strain rate.

We have thus far developed measures of strain and strain rate appropriate for both the spatial and reference configurations. Now we consider appropriate definitions of these quantities for the rotated configuration defined according to the polar decomposition and depicted schematically in Fig. 4.1. This can be done by applying the linear transformation \(\mathbf{R}\) relating the rotated configuration to the spatial one.

The rotated rate of deformation tensor \(\pmb{\mathnormal{D}}\) is thus defined via

(6.16)\[\begin{split}\begin{aligned} \pmb{\mathnormal{D}}(\mathbf{X},t) & = \mathbf{R}^T (\mathbf{X},t) \cdot \mathbf{D}(\varphi(\mathbf{X},t),t) \cdot \mathbf{R}(\mathbf{X},t) \\ {} & = \mathbf{R}^T (\mathbf{D} \circ \varphi) \mathbf{R} . \end{aligned}\end{split}\]

Noting that

(6.17)\[\dot{\mathbf{C}} = 2 \mathbf{F}^T(\mathbf{D} \circ \varphi) \mathbf{F} = 2 \mathbf{U}^T \mathbf{R}^T \mathbf{D} \circ \varphi) \mathbf{R} \mathbf{U} = 2\mathbf{U}^T \pmb{\mathnormal{D}} \mathbf{U} ,\]

we find

(6.18)\[\pmb{\mathnormal{D}} = \frac{1}{2} \mathbf{U}^{-1} \dot{\mathbf{C}} \mathbf{U}^{-1} = \frac{1}{2} \mathbf{C}^{-1/2} \dot{\mathbf{C}} \mathbf{C}^{-1/2} .\]

In connection with the rotated reference, another tensor, \(\pmb{\mathit{L}}\), is sometimes introduced:

(6.19)\[\pmb{\mathnormal{L}} = \dot{\mathbf{R}} \mathbf{R}^{T} .\]

Note that \(\pmb{\mathit{L}}\) is skew:

(6.20)\[\pmb{\mathnormal{L}} + \pmb{\mathnormal{L}}^T = \dot{\mathbf{R}} \mathbf{R}^{T} + \mathbf{R} \dot{\mathbf{R}}^{T} = \frac{\partial}{\partial t} (\mathbf{R}\mathbf{R}^T) = \frac{\partial \mathbf{I}}{\partial t} = 0.\]

We will return later in this document to the various measures associated with the rotated configuration. They have particular importance in the study of material frame indifference.