Continuum mechanics

asdf

"Tensor analysis"

• A 2-tensor $\mathbf{Q}$ can be written as $\mathbf{Q} = \frac{\mathbf{Q} + \mathbf{Q}^\top}{2} + \frac{\mathbf{Q} - \mathbf{Q}^\top}{2}$, where the first operator is symmetric and the second operator is skew-symmetric.
• We also have $\mathbf{Q} = \mathbf{R}\mathbf{S}$ with $\mathbf{R}$ orthogonal and $\mathbf{S}$ symmetric.

The dyadic product:

We define $(\boldsymbol{u} \otimes \boldsymbol{v}) \boldsymbol{w} \equiv (\boldsymbol{w} \cdot \boldsymbol{v}) \boldsymbol{u}$ to be the dyadic product (specific case of tensor product). In a particular basis $\boldsymbol{u} \otimes \boldsymbol{v} = \mathbf{u} \mathbf{v}^\top$.

The natural basis of the space of tensors can be constructed thusly in terms of $\boldsymbol{e}_i \otimes \boldsymbol{e}_j$, for a suitably chosen basis.

We also define the double-product to be $\mathbf{A} : \mathbf{B} \equiv \sum_{i,j=1}^3 A_{ij}B_{ij}$. For higher order tensors this is equivalent to einstein summation notation implicitly specifying summation over the last index.

Kinematics

We assume we have a map $\phi : \mathbb{R}^3 \times \mathbb{R}^+ \to \mathbb{R}^3$ which maps $\boldsymbol{X} \stackrel{\phi(\cdot, t)}{\to} \boldsymbol{x}$

The lagrangian ("material") description of density and the eulerian ("spatial") description can therefore be unified as $\rho = \rho(\boldsymbol{X}, t) = \rho(\phi(\boldsymbol{X}, t), t)$, showing that $\phi$ is identified with the usual "flow map".

The deformation gradient:

Consider two elements of the tangent space of a point $\mathrm{d}\boldsymbol{X}_1, \mathrm{d}\boldsymbol{X}_2$ then the deformation gradient is just the differential of $\phi$ written in coordinates, namely $$\boldsymbol{F} = \frac{\partial \phi_i}{\partial X_j}$$ which has typing $\boldsymbol{F} : T_{\boldsymbol{X}_j} \boldsymbol{X} \to T_{\phi(\boldsymbol{X}_j, t)} \boldsymbol{x}$

We can use the standard argument to write $\mathrm{d}\boldsymbol{x} = \phi_*(\mathrm{d}\boldsymbol{X}) = \boldsymbol{F}\mathrm{d}\boldsymbol{X}$ $\mathrm{d}\boldsymbol{X} = \phi^{-1}_*(\mathrm{d}\boldsymbol{x}) = \boldsymbol{F}^{-1}\mathrm{d}\boldsymbol{x}$

This therefore induces a metric $$\begin{align*} \mathrm{d}\boldsymbol{x}_i \cdot \mathrm{d}\boldsymbol{x}_j &= \boldsymbol{F}\mathrm{d}\boldsymbol{X}_i \cdot \boldsymbol{F}\mathrm{d}\boldsymbol{X}_j \\ &= \mathrm{d}\boldsymbol{X}_i \cdot \boldsymbol{F}^\top\boldsymbol{F}\mathrm{d}\boldsymbol{X}_j \\ &\equiv \mathrm{d}\boldsymbol{X}_i \cdot \boldsymbol{C}\mathrm{d}\boldsymbol{X}_j \\ \end{align*}$$ which can be shown by index juggling. $\boldsymbol{C}$ ends up being called the right Cauchy-Green deformation tensor.

In the other direction, $$\begin{align*} \mathrm{d}\boldsymbol{X}_i \cdot \mathrm{d}\boldsymbol{X}_j &= \mathrm{d}\boldsymbol{x}_i \cdot \boldsymbol{F}^{-\top}\boldsymbol{F}^{-1}\mathrm{d}\boldsymbol{x}_j \end{align*}$$ and the left Cauchy-Green tensor is defined as $\boldsymbol{b}^{-1} = \boldsymbol{F}^{-\top}\boldsymbol{F}^{-1} \implies \boldsymbol{b} = \boldsymbol{F} \boldsymbol{F}^{\top}$

The difference in inner product induced by the flow map can thus be expressed in material form as $\frac{1}{2}\left( \mathrm{d}\boldsymbol{x}_i \cdot \mathrm{d}\boldsymbol{x}_j - \mathrm{d}\boldsymbol{X}_i \cdot \mathrm{d}\boldsymbol{X}_j\right) = \mathrm{d}\boldsymbol{X}_i \cdot \boldsymbol{E} \mathrm{d}\boldsymbol{X}_j$ with $\boldsymbol{E} = \frac{1}{2}(\boldsymbol{C} - \boldsymbol{I})$ and for the spatial form $\frac{1}{2}\left( \mathrm{d}\boldsymbol{x}_i \cdot \mathrm{d}\boldsymbol{x}_j - \mathrm{d}\boldsymbol{X}_i \cdot \mathrm{d}\boldsymbol{X}_j\right) = \mathrm{d}\boldsymbol{x}_i \cdot \boldsymbol{e} \mathrm{d}\boldsymbol{x}_j$ with $\boldsymbol{e} = \frac{1}{2}(\boldsymbol{I} - \boldsymbol{b}^{-1})$

Pushforward/pullback of tensor

For a tensor $\boldsymbol{B} : T_{X_j}\boldsymbol{X} \times T_{X_j}^*\boldsymbol{X} \to \mathbb{R}$ , $\boldsymbol{B}()$ the pushforward to the target is just $\boldsymbol{b}(d\boldsymbol{x}_i, \frac{\partial}{\partial \boldsymbol{x}_j}) \equiv \boldsymbol{B}(\phi_*(d\boldsymbol{x}_i), \phi_*^\top(\frac{\partial}{\partial \boldsymbol{x}_j})$