5 Forces and stresses

This section provides the necessary background on concepts to understand forces and stresses as relevant to this lecture. It constitutes classical content that is covered by most textbooks on continuum mechancis, see our literature list. Here, we are following mostly the corresponding chapters in the book (Gonzalez and Stuart 2001).

5.1 Preliminary considerations

Recall, that we start from the continuum assumption. Based on this, we can identify a material body with a subset $B$ of a Euclidean space $\mathbb R^3$. A material particle is denoted by $\mathbf x \in \mathbb R^3$. $B$ is referred to as the configuration of the body. In general, $B$ will be time dependent.

5.2 Mass density field

Mass is a property that resists to acceleration and is continuously distributed across $B$. Let $vol(B)$ be the volume of $B$:

\[ vol(B) = \int_{\mathbb R^3} \mathcal I_B \, d\mathbf x ,\]

in which $\mathcal I_B$ denotes the characteristic function of $B$. There will be a so-called mass density field (or simply density field or density) $\rho: B \rightarrow \mathbb R$, such that

\[ mass(B) = \int_{B} \rho (\mathbf x) \, d \mathbf x ,\]

and $\rho(\mathbf x) > 0$ for all $\mathbf x \in \mathbb R^3$. For any subset $\Omega \subset B$, we have correspondingly

\[ vol(\Omega) = \int_{\mathbb R^3} \mathcal I_\Omega \, d\mathbf x \quad \text{and} \quad mass(\Omega) = \int_{\Omega} \rho (\mathbf x) \, d\mathbf x .\]

Remark

A formal defintion of $\rho$ is also possible and considers the existence of a limit for $\frac{mass(\Omega)}{vol(\Omega)}$ for regions of vanishing volumes around any arbitrary points $\mathbf x \in B$.

5.3 Center of volume and mass

Later, we will be interested in resulting forces. Often, this requires us to know where the mass center of the center of volume is located for a material at configuration $B$ or any of its subsets $\Omega$. Center of mass and center of volume are defined as first moments according to:

\[\mathbf x_{cov} (\Omega) = \frac{1}{vol(\Omega)} \int_{\mathbb R^3} \mathbf x \mathcal I_B \, d \mathbf x\]

and \[\mathbf x_{com} (\Omega) = \frac{1}{mass(\Omega)} \int_{\mathbb R^3} \mathbf x \rho (\mathbf x) \, d \mathbf x\]

Remark

$\mathbf x_{cov}$ and $\mathbf x_{com}$ can be inside or outside of $\Omega$ or even $B$.

In the next two sections, we will discuss forces that denote the mechanical interaction between part of a body or the body and its environment. We distinguish

Body forces that are exerted at interior points of a material at configuration $B$ and typically arise from non-contact action at a distance, and
Surface forces that are either exerted on internal surfaces between seperate parts of a material, or on external surfaces between the body and its environment. Surface forces are typically of contact-type.

5.4 Body forces

One very important example are body forces due to gravitational acceleration, see also last section on the incompressible Euler model. A formalization of the body force field per unit volume acting on $B$ reads

\[ \widehat{\mathbf b} : B \rightarrow \mathcal V\]

The resultant force on $\Omega$ is then given as a single vector and can be computed via

\[ \mathbf r_b (\Omega) = \int_\Omega \widehat{\mathbf b} (\mathbf x) \, d \mathbf x.\]

The resultant torque on $\Omega$ is correspondingly given by

\[ \mathbf \tau_b (\Omega) = \int_\Omega \left( \mathbf x - \mathbf z \right) \times \widehat{\mathbf b} (\mathbf x) \, d \mathbf x.\]

ALternatively, the body force field is often given as the body force field per unit mass $ b : B V$, defined as

\[ \mathbf b (\mathbf x): = \rho(\mathbf x)^{-1} \widehat{\mathbf b} (\mathbf x),\]

in which case resultant force and torque are given by

\[ \mathbf r_b (\Omega) = \int_\Omega \rho (\mathbf x) \mathbf b (\mathbf x) \, d \mathbf x \quad \text{and} \quad \mathbf \tau_b (\Omega) = \int_\Omega \left( \mathbf x - \mathbf z \right) \times \rho(\mathbf x) \mathbf b (\mathbf x) \, d \mathbf x.\]

Remark

The resultant torque can be determined for any reference point $\mathbf z$. We can also pick a reference point $\mathbf z$, such that the torque vanishes.

5.5 Surface forces

Whenever we speak of an internal surface force, we mean the surface force along an imaginary surface within the interior of configuration $B$. Let’s assume an internal surface to be given by $\Gamma \subset B$. The internal surface separates a positive part of $B$ from a negative part of $B$. Surface $\Gamma$ has unit normals given by

\[ \mathbf n : \Gamma \rightarrow \mathcal N \subset \mathcal V,\]

in which $\mathcal N$ stands for all vectors of unit length, hence $|\mathbf n| = 1$. The hat indicates that the unit vectors are chosen in a way, such that they point into the positive sides. If $\Gamma$ denotes an external surface, hence the outer boundary $\partial B$ of material at configuration $B$, the orientation of $ $ is chosen such that it points in outward direction.

The force per unit area exerted by

material on the positive side upon material to the negative side (internal surfaces), or by
material on the ambient on the body (external surface)
is referred to as traction or surface force field for $\Gamma$. It is defined as

\[ \mathbf t_{\widehat{\mathbf n}} : \Gamma \rightarrow \mathcal V. \]

Resultat force and torque due to a traction field as then given by

\[ \mathbf r_s (\Gamma) = \int_\Gamma t_{\widehat{\mathbf n}} \, d \mathbf \sigma \quad \text{and} \quad \mathbf \tau_s (\Gamma) = \int_\Gamma \left( \mathbf x - \mathbf z \right) \times t_{\widehat{\mathbf n}} \, d \mathbf \sigma.\]

Note, that in contrast to the previously introduced resultant force and torque due to the body force field, we are here considering surface integrals as we are averaging along a surface.

5.5.1 Cauchy’s postulate

The traction field $t_{\widehat{\mathbf n}}$ on a surface $\Gamma$ in $B$ depends only on $\mathbf x$ and $\widehat{\mathbf n}$, such that there is a function

\[ \mathbf t : \mathcal N \times B \rightarrow \mathcal V,\]

with $t_{\widehat{\mathbf n}} = t(\widehat{\mathbf n}(\mathbf x),\mathbf x)$. $\mathbf t$ is called the traction function for $B$.

Remark

Note, that the traction function is a function of the configuration $B$ and requires not defintion of complete surfaces $\Gamma$. It is hence local in the sense that a location $\mathbf x$ and a local orientation $\widehat{\mathbf n}$ have to be specified. There is nofurther dependence, for instance on the local curvature of the surface $\nabla \widehat{\mathbf n}$.

5.5.2 Law of Action and Reaction

This law is a direct consequence of Cauchy’s postulate. Is assumes continuity of the traction field $\mathbf t(\mathbf n, \mathbf x)$ and reads

\[ \mathbf t(-\mathbf n, \mathbf x) = - \mathbf t(\mathbf n, \mathbf x),\]

meaning that the traction associated with the mirrored surface corresponds to the negative traction of the original surface. In mathematical terms, this can be understood as skew-symmetry of the traction function in the first argument. For a proof of this, we refer to the book (Gonzalez and Stuart 2001).

Remark

The law of Action and Reaction allows us to infer directly on forces acting on the mirrored surface. Knowing the traction on one side, hence immediately tells us the traction on the other side. It is hence no longer necessary to strictly differentiate the orientation, such that we can let go of the hat from now on and write $\mathbf n$ always. We still stick to our convention of demanding normals of the external bounding surface $\partial B$ to point in the outward direction.

5.5.3 Cauchy’s Theorem or Existence of a stress tensor

Let $\mathbf t : \mathcal N \times B \rightarrow \mathcal V$ be a traction function for $B$. $\mathbf t$ is then linear in $\mathbf n$. Hence, for each $\mathbf x \in B$ there is a second order tensor $\mathbf S \in \mathcal V^2$, such that

\[ \mathbf t(\mathbf n, \mathbf x) = \mathbf S (\mathbf x) \mathbf n.\]

$\mathbf S : B \rightarrow \mathcal V^2$ is called the Cauchy stress field for $B$. A proof can once again be found in the book (Gonzalez and Stuart 2001).

Remark

Note, that the following version all stand for the same

$\mathbf t(\mathbf n(\mathbf x), \mathbf x) = \mathbf S (\mathbf x) \mathbf n (\mathbf x)$
$\mathbf t(\mathbf x) = \mathbf S (\mathbf x) \mathbf n$
$\mathbf t = \mathbf S \mathbf n$
$t_i = S_{ij} n_j$

yet make dependencies on position $\mathbf x$ and local orientation $\mathbf n$ more or less explicit.

The nine components of $\mathbf S( \mathbf x)$ can be understood as the components of the three traction vectors $\mathbf t(\mathbf e_j, \mathbf x)$ on the coordinate planes at $\mathbf x$.

5.5.4 Equilibrium of forces

A body is said to be in an equilibrium configuration, if resultant forces and torques vanish for every $\Omega \subset B$, which reads

\[\mathbf r (\Omega) = \int_\Omega \mathbf x \times \rho(\mathbf x) \mathbf b(\mathbf x) \, d \mathbf x + \int_{\partial \Omega} \mathbf x \times \mathbf t (\mathbf x) \, d \sigma = 0\]

and

\[\mathbf \tau (\Omega) = \int_\Omega \rho(\mathbf x) \mathbf b(\mathbf x) \, d \mathbf x + \int_{\partial \Omega} \mathbf t (\mathbf x) \, d \sigma = 0.\]

These global type equilibrium conditions can be shown to translate to local relations:

\[\left( \nabla \cdot \mathbf S \right) (\mathbf x) + \rho(\mathbf x) \mathbf b(\mathbf x) = 0\]

and

\[\mathbf S^T (\mathbf x) = \mathbf S (\mathbf x),\]

in which $\mathbf S$ is the Cauchy stress tensor. A proof of the first relation is immediately obvious, when exploiting the existence of a Cauchy tensor and applying divergence theorem. The second implication is less obvious, and we refer the reader again to the book (Gonzalez and Stuart 2001).

5.5.5 Fundamental stress states

i) Spherical / Eulerian stress state

\[\mathbf S = - p \mathbf I,\]

in which $p$ is a scalar. In this case, the traction on any surface with normal $\mathbf n$ at $\mathbf x$ is given by

\[\mathbf t = \mathbf S \mathbf n = - p \mathbf n.\]

It is hence always normal to the surface, regardless which one it is.

ii) Uniaxial stress state

This stress state is given, if there exists a unit vector $\mathbf n_a \in \mathcal N$ and a scalar $\alpha \in \mathbb R$, such that

\[ \mathbf S = \alpha \, \mathbf n_a \otimes \mathbf n_a.\]

We refer to a state in which $\alpha > 0$ as pure tension and a state in which $\alpha < 0$ as pure compression. We hence get

\[\mathbf t = \mathbf S \mathbf n = (\mathbf n_a \cdot \mathbf n) \alpha \mathbf n_a,\]

in which the first term corresponds to a projection. Traction is always parallel to $\mathbf n_a$.

iii) Pure shear stress state

This stress state is given, if there exist orthogonal unit vectors $\mathbf n_a, \mathbf n_b \in \mathcal N$ and a scalar $\tau \in \mathbb R$, such that

\[ \mathbf S = \tau \, \left( \mathbf n_a \otimes \mathbf n_b + \mathbf n_b \otimes \mathbf n_a \right).\]

The traction results in

\[\mathbf t = \mathbf S \mathbf n = (\mathbf n_a \cdot \mathbf n) \tau \mathbf n_b + (\mathbf n_b \cdot \mathbf n) \tau \mathbf n_a.\]

Hence, $\mathbf t = \tau \mathbf n_a$ if $\mathbf n = \mathbf n_b$ and $\mathbf t = \tau \mathbf n_b$ if $\mathbf n = \mathbf n_a$.

5.5.6 Principal, normal and shear stresses

$\mathbf S$ is symmetric and invertible. It hence has three eigenvalues and three mutually exclusive, orthogonal eigenvectors:

\[\underbrace{\mathbf t}_{\text{associate traction}} = \mathbf S \underbrace{\mathbf e}_{\text{principle stress direction}} = \underbrace{\sigma}_{\text{principle stress}} \mathbf e.\]

In order to interpret these, we at first consider quite generally, how we could decompose any traction $\mathbf t$ at location $\mathbf x$. One possibility is to single out the part pointing into normal direction, hence the normal traction

\[\mathbf t_n = \left(\mathbf t \cdot \mathbf n \right) \mathbf n\]

and the remainder denoting the shear traction

\[\mathbf t_s = \mathbf t - \left(\mathbf t \cdot \mathbf n \right) \mathbf n.\]

Adding up both yields to original traction vector:

\[\mathbf t = \mathbf t_s + \mathbf t_n\]

We correspondingly identify normal stress $\sigma_n := |\mathbf t_n|$ and shear stress $\sigma_s := |\mathbf t_s|$ on surface $\mathbf n$ at location $\mathbf x$.

If now $\mathbf n$ corresponds to the principle stress directions, hence $\mathbf n = \mathbf e$ (see above), we get

\[\mathbf t_n = \left(\mathbf t \cdot \mathbf e \right) \mathbf e = \sigma \mathbf e,\]

in which case $\sigma_n = |\sigma|$. On the other hand, we get

\[\mathbf t_s = \mathbf t - \left(\mathbf t \cdot \mathbf e \right) \mathbf e = 0,\]

hence $\sigma_s = 0$. This means that the shear stress on a surface is zero if the surface normal $\mathbf n$ corresponds to a principle stress directions (and vice versa).

5.5.7 Decomposition of the stress tensor

At any point $\mathbf x$ in a continuum body at configuration $B$, we can decompose the Cauchy stress tensor $\mathbf \sigma$ into the sum of two parts. These are (1) the spherical stress tensor $\mathbf S_s := -p \mathbf I$, and (2) the deviatoric part $\mathbf S_d := \mathbf \sigma + p \mathbf I$,

such that $\mathbf \sigma = \mathbf S_s + \mathbf S_d$, and

\[ p:= - \frac{1}{3} \left( \sigma_1 + \sigma_2 + \sigma_3 \right) = tr(\mathbf \sigma)\]

is referred to as the pressure, given by the arithmetic mean of the principle stresses, which is also an invariant of the stress tensor. It is positive in compressive states.

$\mathbf S_s$ is a diagonal second order tensor and denotes the part of the overall stress that tends to change the volume of the body without changing the shape.

$\mathbf S_d$ is the part that complements to $\mathbf S$ and decodes any shear stress contribution. It tends to change the shape without changing the volume.