Samer Adeeb

Displacement and Strain: Material Time Derivative

Material Time Derivative

For time dependent deformations, the material time derivative, is the derivative of scalar, vector or tensor fields as a function of the material point with respect to time. Let \Omega_0 and \Omega represent the reference and the deformed configurations of a body embedded in \mathbb{R}^3. Let x(X,t)\in \Omega represent the image of the vector X\in\Omega_0 at a certain point in time t\in\mathbb{R}. Let v and L and be the velocity and the velocity gradient fields. Let \phi:\Omega\times\mathbb{R}\rightarrow \mathbb{R}, u:\Omega\times\mathbb{R}\rightarrow \mathbb{R}^3 and T:\Omega\times\mathbb{R}\rightarrow \mathbb{M}^3 be a scalar, a vector and a tensor field respectively. I.e., \phi=\phi(x,t), u=u(x,t) and T=T(x,t). Then, the material time derivative of these fields, denoted \frac{da}{dt} with a being \phi, u or T is the total derivative of these quantities as follows:

    \[ \begin{split} &\frac{d\phi}{dt}=\dot{\phi}=\frac{\partial\phi(x,t)}{\partial t}+\frac{\partial\phi(x,t)}{\partial x}\cdot \frac{\partial x}{\partial t}=\frac{\partial\phi(x,t)}{\partial t}+\mbox{grad}(\phi)\cdot v\\ &\frac{du}{dt}=\dot{u}=\frac{\partial u(x,t)}{\partial t}+\frac{\partial u(x,t)}{\partial x}\frac{\partial x}{\partial t}=\frac{\partial\phi(x,t)}{\partial t}+\mbox{grad}u(v)\\ &\frac{dT}{dt}=\dot{T}=\frac{\partial T(x,t)}{\partial t}+\frac{\partial T(x,t)}{\partial x}:\frac{\partial x}{\partial t} \end{split} \]

Notice that \dot{\phi} is a scalar valued field, \dot{u} is a vector valued field, while \dot{T} is a tensor valued field. The components of these quantities are given by:

    \[ \begin{split} &\dot{\phi}=\frac{\partial\phi(x,t)}{\partial t}+\sum_{i=1}^3\frac{\partial\phi(x,t)}{\partial x_i} \frac{\partial x_i}{\partial t}\\ &\dot{u}_i=\frac{\partial u_i(x,t)}{\partial t}+\sum_{j=1}^3\frac{\partial u_i(x,t)}{\partial x_j}\frac{\partial x_j}{\partial t}\\ &\dot{T}_{ij}=\frac{\partial T_{ij}(x,t)}{\partial t}+\sum_{k=1}^3\frac{\partial T_{ij}(x,t)}{\partial x_k}\frac{\partial x_k}{\partial t} \end{split} \]

In case where the fields are functions of the reference configuration, i.e., \phi:\Omega_0\times\mathbb{R}\rightarrow \mathbb{R}, u:\Omega_0\times\mathbb{R}\rightarrow \mathbb{R}^3 and T:\Omega_0\times\mathbb{R}\rightarrow \mathbb{M}^3 are the respective scalar, vector and tensor fields, then, the material time derivative of these fields, denoted \frac{Da}{Dt} with a being \phi, u or T is the total derivative of these quantities. However, since X is fixed, the only variable that appears in the derivation is time:

    \[ \begin{split} &\frac{D\phi}{Dt}=\frac{\partial\phi(X,t)}{\partial t}\\ &\frac{Du}{Dt}=\frac{\partial u(X,t)}{\partial t}\\ &\frac{dT}{dt}=\frac{\partial T(X,t)}{\partial t} \end{split} \]

And the components of these quantities are given by:

    \[ \begin{split} &\frac{D\phi}{Dt}=\frac{\partial\phi(X,t)}{\partial t}\\ &\left(\frac{Du}{Dt}\right)_i=\frac{\partial u_i(X,t)}{\partial t}\\ &\left(\frac{dT}{dt}\right)_{ij}=\frac{\partial T_{ij}(X,t)}{\partial t} \end{split} \]

Example:

In the following example, the scalar field temperature is used to illustrate the above concepts. Assuming a plate of width 4 units and height of 2 units is rotating around the origin with a speed of \omega=2\pi/1000 rad/second. If the spatial distribution of the temperature in the medium is given by Temp=70(1-0.2x_1)(1-0.1x_2) units of temperature. If we assume that the speed of the plate is slow enough such that the plate assumes the temperature of the medium, then the material time derivative of the temperature is non zero.
The position function at time t is equal to:

    \[ x=QX= \left(\begin{array}{cc} \cos{\omega t} & -\sin{\omega t}\\ \sin{\omega t} & \cos{\omega t} \end{array} \right) \left(\begin{array}{c} X1\\ X2 \end{array} \right) \]

where \omega=2\pi/1000
The spatial velocity field is given by the following vector function:

    \[ v=\dot{x}=\dot{Q}Q^{-1}x=\left( \begin{array}{c} -\pi x_2/500 \\ -\pi x_1/500 \end{array} \right) \]

The spatial gradient of temperature field is given by the following vector function:

    \[ \mbox{grad}(Temp)=\left(\begin{array}{c} -14(1-0.1x_2)\\ -7(1-0.2x_1) \end{array} \right) \]

The material time derivative of the temperature is given by the following scalar function:

    \[ \frac{d(Temp)}{dt}=0+\mbox{grad}(Temp)\cdot v \]

The following tool draws the velocity vector field, the spatial gradient of temperature vector field and the contour plot of the material time derivative of the temperature. Change the value of t between 0 and 1000 units to see the effect on the position of the plate and the calculated fields.

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