2 Fundamentals
2.1 Continuum Assumption
In order to model the motion, deformation and phase-change of a material, we think of it as an idealized medium that is continuously filling a region in three-dimensional space. We discriminate a particle point and a material point:
material point: the “exact” position within the medium with its micro-scale properties
particle point: an intermediate-scale position within the medium that has all its properties
Conclusion: The particle point should be much larger than a material point, yet from a continuum mechanical perspective, a particle point should be infinitesimally small. How can we formalize this concept?
Schematic presented in class.
Intermediate asymptotics exists for materials, in which the characteristic length-scale of the material’s microstructure \(X_1\) (often also denoted \(\delta\)), is much smaller than the characteristic length scale of the problem setting \(X_2\) (often also denoted by \(L\)), hence (\(\delta \lll L\)).
In such a situations, we can safely assume the continuum assumption to apply within an intermediate interval \(\mathcal I\).
Examples are:
Fluids: The ratio of the molecular free path and the characteristic length scale is usually very small. This ratio is also referred to as the Knudsen Number. Systems with a Knudsen Number \(<0.1\) can be regarded as continuous.
Porous media: Porous media can be treated as continuous, when we are interested in a scale that is much larger than the characteristic pore size, e.g. in reservoir engineering. This is different, when we are interested in processes on the pore size level.
2.2 Observer
An observer of a continuous medium provides
a reference system, e.g. an inertial reference system
a clock that allows to measure time
a system of units sufficient for the phenomena that we consider
2.3 Tensor algebra in a nutshell
The content of this subsection is a concise selection of chapters 1 and 2 in (Gonzalez and Stuart 2001).
Unless otherwise stated, we refer to space as a Euclidean 3d space denoted by \(\mathbb R^3\), in which elements of \(\mathbb R\) are scalars. Elements in \(\mathbb R^3\) are points in space and can be written in coordinates \(\mathbf x = (x_1,x_2,x_3)^T\).
2.3.1 Vectors
A vector is an element of a vector space. Here, we think of a vector as an element of \(\mathbb R^3\) that has a magnitude and a direction, hence
\[ \mathbf v = (v_1,v_2,v_3)^T = \underbrace{|\mathbf v|}_{\text{magnitude}} \quad \underbrace{\mathbf e_v}_{\text{direction}}\]
We have always \(|\mathbf v| \leq 0\). \(\mathbf v\) with \(|\mathbf v|=1\) is called a unit vector. Two vectors are the same if they coincide in magnitude and direction (regardless of any specific position, in which they might be drawn).
Per the structure of a vector space, we have for all \(\mathbf u, \mathbf v \in \mathcal V\) and \(\alpha \in \mathcal V\), hence \[\mathbf u + \mathbf v \in \mathcal V \qquad \alpha \mathbf u \in \mathcal V\]
For a basis \(\{\mathbf e_1, \mathbf e_2, \mathbf e_3 \}\), we get can write any vector in its components according to
\[ \mathbf v = v_1 \mathbf e_1 + v_2 \mathbf e_2 + v_3 \mathbf e_3 \quad \left( = v_i \mathbf e_i \right)\]
with \([\mathbf v] = (v_1,v_2,v_3)^T\) being the coordinates of vector \(\mathbf v\).
2.4 Second order tensors
Physical quantities, such as stress and strain cannot be represented as vectors, but rather by linear transformations between vectors. A second order tensor on the vector space \(\mathcal V\) is hence a mapping \(\mathbf T : \mathcal V \rightarrow \mathcal V\), which is linear for all \(\mathbf u, \mathbf v \in \mathcal V\) and \(\alpha \in \mathcal V\), hence
\[ \mathbf T(\mathbf u + \mathbf v) = \mathbf T(\mathbf u) + \mathbf T(\mathbf v) \quad \mathbf T( \alpha \mathbf u) = \alpha \mathbf T(\mathbf u)\]
For a basis \(\{\mathbf e_1, \mathbf e_2, \mathbf e_3 \}\), we get can write any second order tensor in its components according to
\[ [S] = S_{ij} = \mathbf e_i \cdot \mathbf S \mathbf e_j\]
The coordinates can hence be represented as a matrix \(S_{ij}\).
The space of second order tensors is itself a vector space of basis \(\{\mathbf e_i \otimes \mathbf e_j\}\). Note, that the dyadic product is defined as
\[(\mathbf u \otimes \mathbf v) \mathbf a = (\mathbf a \cdot \mathbf v)\mathbf u\]
for all \(\mathbf a \in \mathcal V\). We then get
\[ [S + T] = [S] + [T] \qquad [\alpha S] = \alpha [T]\]
A second order tensor can always be decomposed into a symmetric and a skewsymmetric part \[ \mathbf S = \mathbf E + \mathbf W\]
in which \[ \mathbf E = \frac{1}{2} (\mathbf S + \mathbf S^T) \qquad \mathbf W = \frac{1}{2} (\mathbf S - \mathbf S^T)\]
As mappings, \(\mathbf E\) and \(\mathbf W\) are referred to as \(sym(\mathbf S)\) and \(skew(\mathbf S)\).
A skew symmetric tensor can furthermore always be expressed as a vector product, in the sense that there exist a \(\mathbf w\), such that
\[\mathbf W \mathbf v = \mathbf w \times \mathbf v\]
\(\mathbf w\) is also denoted by \(vec(\mathbf W)\) and referred to as the axial vector.
2.5 Scalar, vector and tensor fields
Under the continuum assumption, we can assume that we can describe the state of the system in terms of continuous fields. The fields are functions of position \(\mathbf x = (x,y,z)^T\) and time \(t\).
Relevant to this lecture will be: \[ \begin{aligned} &\rho(\mathbf x,t) : \text{density field (mass per unit volume)}\\ &\mathbf v (\mathbf x,t) = (v_1(\mathbf x,t),v_2(\mathbf x,t),v_2(\mathbf x,t))^T : \text{velocity field}\\ &p(\mathbf x,t) : \text{pressure field}\\ &T(\mathbf x,t) : \text{temperature field }\\ &\mathbf \sigma(\mathbf x,t) : \text{stress field}\\ \end{aligned} \]
2.6 Index and vector notation
As an element of a vector space, a vector can always be written in its basis:
\[ \mathbf{v} \left( \mathbf{x},t \right) = u \left( \mathbf{x},t \right) \mathbf{e}_x + v \left( \mathbf{x},t \right)\mathbf{e}_y + w \left( \mathbf{x},t \right) \mathbf{e}_z \]
Alternatively, basis and vector components can be numbered, which yields
\[ \mathbf{v} = v_1 e_1 + v_2 e_2 + v_3 e_3 = \sum_{i=1}^3 v_i e_i \]
If an index appears twice in a term, summation is assumed
\[ \mathbf{v} = \sum_{i=1}^3 v_i e_i = v_i e_i \]
A scalar product between vector \(\mathbf{v}\) and vector \(\mathbf{w}\) can then concisely be written as
\[ \mathbf{v} \cdot \mathbf{w} = v_i w_i \]
2.7 Gradient, divergence, Laplacian
All of the following statements implicitly imply differentiability.
The gradient of a scalar field \(\phi: \mathbb R^3 \rightarrow \mathbb R\) is a vector field given by
\[ \nabla \phi (\mathbf x) = \frac{\partial \phi}{\partial x_i} (\mathbf x) \mathbf e_i\]
The gradient of a vector field \(\mathbf v: \mathbb R^3 \rightarrow \mathbb R^3\) is a second order tensor field given by
\[ \nabla \mathbf v (\mathbf x) = \frac{\partial v_i}{\partial x_j} (\mathbf x) \mathbf e_i \otimes \mathbf e_j\]
The divergence of a vector field \(\mathbf v: \mathbb R^3 \rightarrow \mathbb R^3\) is a scalar field given by
\[ \nabla \cdot \mathbf v = tr( \nabla \mathbf v) \qquad (\nabla \cdot \mathbf v) (\mathbf x) = \frac{\partial v_i}{\partial x_i} (\mathbf x) \]
The Laplacians of a scalar field \(\phi: \mathbb R^3 \rightarrow \mathbb R\) and a vector field \(\mathbf v: \mathbb R^3 \rightarrow \mathbb R^3\) are defined as
\[ \triangle \phi = \nabla \cdot ( \nabla \phi) \qquad \triangle \mathbf v = \nabla \cdot ( \nabla \mathbf v)\]
2.8 Local, convective and material derivative
The aforementioned form of the state variables is called the Eulerian formulation. In contrast to this, we can look at a material particle initially located at \(\mathbf x_0\) and track its trajectory during motion:
\[ \mathcal{\mathbf X} (\mathbf x_0;t), \quad \mathcal{\mathbf X}:\mathbb R^+ \rightarrow \mathbb R^d,\quad \text{with} \quad \mathcal{\mathbf X} (\mathbf x_0;t) = \left(X_1(\mathbf x_0;t),X_2(\mathbf x_0;t),X_3(\mathbf x_0;t) \right) \]
Here, \(\mathbf x_0\) denotes that the trajectory starts at \(\mathbf x_0\) at time zero. It is, hence, no real space coordinate and can rather be seen as a parameter of \(\mathcal{\mathbf X}\). We will therefore drop \(\mathbf x_0\) for the rest of this section and refer to the trajectory as
\[ \mathcal{\mathbf X} (t) = (X_1(t),X_2(t),X_3(t)). \]
Tracking the evolution of the density along the trajectory of a fluid parcel, hence \(\rho(\mathcal{\mathbf X}(t),t)\), or shorter \(\rho(\mathcal{\mathbf X},t)\), is referred to as the density in Lagrangian formulation. Let’s look at the total density \(\rho(\mathcal{\mathbf X},t)\) change along the trajectory:
\[ \begin{aligned} \frac{d}{dt} \rho(\mathcal{\mathbf X},t) &= \frac{d}{dt} \rho(X_1(t),X_2(t),X_3(t),t) \\[1em] & = \underbrace{\frac{d}{dt} X_1(t)}_{=v_1(\mathbf x,t)} \cdot \partial_x \rho +\underbrace{\frac{d}{dt} X_2(t)}_{=v_2(\mathbf x,t)} \cdot \partial_y \rho +\underbrace{\frac{d}{dt} X_3(t)}_{=v_3(\mathbf x,t)} \cdot \partial_z \rho +\underbrace{\frac{d}{dt} t}_{=1} \cdot \partial_t \rho &\\[1em] & = \partial_t \rho + v_1 \cdot \partial_{x_1} \rho + v_2 \cdot \partial_{x_2} \rho + v_3 \cdot \partial_{x_3} \rho \\[2em] & = \underbrace{\underbrace{\partial_t \rho}_{\text{local/time derivative}} + \underbrace{ \mathbf v \cdot \nabla \rho }_{\text{convective derivative}}}_{\text{material/total derivative denoted by } \frac{D}{Dt}} \end{aligned} \]
The resulting relation between the total derivative and its local and convective counterparts reads:
\[ \begin{equation*} \frac{D}{Dt} \rho = \partial_t \rho + \mathbf v \cdot \nabla \rho \end{equation*} \]
The Lagrangian notion has been introduced to understand the difference between total, local and convective derivation. Unless otherwise stated, we will work with the Eulerian formulation from now.