5  Forces and stresses

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

5.1 Preliminary considerations

Recall that we started from the continuum assumption. Based on this we identify a material body with a subset \(B\) of Euclidean space \(\mathbb R^3\). A material particle can be located at coordinate \(\mathbf x \in \mathbb R^3\). \(B\) 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 \(B\) defined as

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

in which \(\mathcal I_B : \mathbb R^3 \rightarrow \{0,1\}\) denotes the characteristic function of \(B\). There is a so-called mass density field, or simply density field or density \(\rho: B \rightarrow \mathbb R\), for which \(\rho(\mathbf x) > 0\) for all \(\mathbf x \in \mathbb R^3\). Mass \(mass(B)\) of a body at configuration \(B\) results from the volume integral \[ mass(B) = \int_{B} \rho (\mathbf x) \, d \mathbf x.\]

For any subset \(\Omega \subset B\), we have

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

and

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

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

Eventually, we are interested in resulting forces acting on a body at configuration \(B\) or a subset \(\Omega\) thereof. This requires us to know the location of its center of mass and its center of volume. Both 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 \[\mathbf x_{com} (\Omega) = \frac{1}{mass(\Omega)} \int_{B} \mathbf x \rho (\mathbf x) \, d \mathbf x\]

Note, that \(\mathbf x_{cov}\) and \(\mathbf x_{com}\) can both be within 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

The body force field per unit volume acting a body at configuration \(B\) reads

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

in which \(\mathcal V\) denotes a vector space. The resultant force acting on \(B\) is given by a single vector and can be computed via

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

The resultant torque acting on \(B\) is correspondingly given by

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

The body force field per unit mass \(\mathbf b : B \rightarrow \mathcal V\) is defined as

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

in which case resultant force and torque are given by

\[ \mathbf r_b (B) = \int_B \rho (\mathbf x) \mathbf b (\mathbf x) \, d \mathbf x\]

and

\[\mathbf \tau_b (B) = \int_B \left( \mathbf x - \mathbf z \right) \times \rho(\mathbf x) \mathbf b (\mathbf x) \, d \mathbf x.\]

The resultant torque can be determined for any reference point \(\mathbf z\). Note, that we can pick a reference point \(\mathbf z\), such that the torque vanishes. Note also, that all definitions in this section carry over to subset \(\Omega \subset B\).

5.5 Surface forces

Whenever we speak of an internal surface force, we mean the surface force acting on 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

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

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

A traction or surface force field for \(\Gamma\) refers to a force per unit area exerted by

  • material on the positive side of \(B\) upon material to the negative side (internal surfaces), or by
  • material on the ambient on the body \(B\) on material on the outer surface of \(B\) (external surface)

The traction denotes a vector. It is defined as

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

Resultat force and torque due to a traction field acting on surface \(\Gamma\) are hence defined as

\[ \mathbf r_s (\Gamma) = \int_\Gamma t_{\widehat{\mathbf n}} \, d \mathbf \sigma \]

and

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

In contrast to the previous section on body forces, 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 local orientation \(\widehat{\mathbf n}\) and position \(\mathbf x\), such that there is a mapping

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

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

Quite remarkably, the traction function is a function of \(B\) alone and requires no defintion of complete surfaces \(\Gamma\). It is local and has no dependence beyond position \(\mathbf x\) and orientation \(\widehat{\mathbf n}(\mathbf x)\). It for instance does not depend on the local curvature of the surface \(\nabla \widehat{\mathbf n}(\mathbf x)\).

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 acting on the original surface. In mathematical terms, this can be understood as skew-symmetry of the traction function in the first argument. For a proof 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 immediately tells us the traction on the other side. It means that we no longer have todistinguish between positive and negative orientation and can let go of the hat notation, hence write \(\mathbf n\) always. We will 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 \(B\). The traction function \(\mathbf t\) 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 again be found in the book (Gonzalez and Stuart 2001).

The following versions 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 \(\mathbf x\).

5.6 Equilibrium of forces

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

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

and

\[\mathbf \tau (\Omega) = \underbrace{\int_\Omega (\mathbf x - \mathbf z) \times \rho(\mathbf x) \mathbf b(\mathbf x) \, d \mathbf x}_{= \mathbf \tau_b (\Omega)} + \underbrace{\int_{\partial \Omega} (\mathbf x - \mathbf z) \times \mathbf t (\mathbf x) \, d \sigma = 0}_{= \mathbf \tau_s (\Omega)}.\]

These global type equilibrium conditions translate into 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 immediately follows from substituting in the Cauchy tensor and applying divergence theorem. The second implication is less obvious. A nice summary is given in (Gonzalez and Stuart 2001).

5.7 Fundamental stress states

5.7.1 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 its orientation.

5.7.2 Uniaxial stress state

This stress state implies the existence of 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, and get

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

Here, the first term corresponds to a projection. Traction is always parallel to \(\mathbf n_a\).

5.7.3 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 \(\mathbf n = \mathbf n_a\).

5.8 Principal, normal and shear stresses

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

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

In order to interpret these, we at first consider a quite general decomposition of traction \(\mathbf t\) at location \(\mathbf x\).

Given a normal vector \(\mathbf n\), we can 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

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

Per construction, the sum of both yields to original traction vector:

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

We correspondingly define normal stress

\[\sigma_n := |\mathbf t_n|\]

and shear stress

\[\sigma_s := |\mathbf t_s|\]

on surface \(\mathbf n\) at location \(\mathbf x\).

If in this configuration, normal vector \(\mathbf n\) corresponds to the principle stress directions, hence \(\mathbf n = \mathbf e\) as 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 vanishes, if the surface normal \(\mathbf n\) corresponds to one of the principle stress directions.

5.8.1 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

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

\(p\) is referred to as the pressure, given by the arithmetic mean of the principle stresses. Since \(p = tr(\sigma)\) it is also an invariant of the stress tensor. We’ll define it to be 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.