4. Stress

Stress Vector

$$\vec{t}=\frac{d{\vec{F}}_n}{dA}=\left[\begin{array}{c}{\sigma }_{nn}\\ {\tau }_{nm}\\ {\tau }_{np}\end{array}\right]$$

Where $\sigma$ is the normal stress and $\tau$ is tangential stress. The coordinate space is normal-space.

Stress components have two indices (subscripts), ${\sigma }_{ii},{\tau }_{ik}$:

• The first is the axis of the plane cutting the area
• The second is the direction of the force

Stress Tensor

Reciprocity of stress

$$\begin{array}{rlrlr}{\tau }_{xy}={\tau }_{yx}& & {\tau }_{xz}={\tau }_{zx}& & {\tau }_{yz}={\tau }_{zy}\end{array}$$

Therefore, we only have 6 total stress components in an infinitesimal volume. (Including ${\sigma }_{xx},{\sigma }_{yy},{\sigma }_{zz}$) The above is true in statics, because if it wasn’t then we would get movement since net forces would not be zero.

$$\vec{t}=[\sigma ]\vec{n}$$

Where:

$$[\sigma ]=\left[\begin{array}{ccc}{\sigma }_{xx}& {\tau }_{xy}& {\tau }_{xz}\\ {\tau }_{xy}& {\sigma }_{yy}& {\tau }_{yz}\\ {\tau }_{xz}& {\tau }_{zy}& {\sigma }_{zz}\end{array}\right]$$

Principal Stresses

A principal stress $\lambda$ is one such that:

$$\vec{t}=[\sigma ]\vec{n}=\lambda \vec{n}$$

Recalling from Linear Algebra and Geometry, we can notice that principal stresses are eigenvalues of the stress tensor. We can therefore find them by using the formula:

$$\det ([\sigma ]-\lambda I)=0$$

Where $I$ is the identity matrix. The associated eigenvectors are the principal directions.

In the principal frame of reference, the stress tensor becomes:

$$[\sigma ]=\left[\begin{array}{ccc}{\sigma }_1& 0& 0\\ 0& {\sigma }_2& 0\\ 0& 0& {\sigma }_3\end{array}\right]$$

Where ${\sigma }_1,{\sigma }_2{\sigma }_3$ are principal stresses (previously $\lambda$).

Note

For all stress tensors, principal stresses exist, since $[\sigma ]$ is symmetrical.

We can use Mohr’s Circle to calculate components of the stress tensor.

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$$\left[\begin{array}{ccc}{\sigma }_{xx}& {\tau }_{xy}& {\tau }_{xz}\\ {\tau }_{xy}& {\sigma }_{yy}& {\tau }_{yz}\\ {\tau }_{xz}& {\tau }_{yz}& {\sigma }_{zz}\end{array}\right]=\underset{\begin{array}{c}\text{Isotropic}\\ \text{(Volume changes)}\end{array}}{\underbrace{\left[\begin{array}{ccc}{\sigma }_m& 0& 0\\ 0& {\sigma }_m& 0\\ 0& 0& {\sigma }_m\end{array}\right]}}+\underset{\begin{array}{c}\text{Deviatoric}\\ \text{(Shape changes)}\end{array}}{\underbrace{\left[\begin{array}{ccc}{\sigma }_{xx}-{\sigma }_m& {\tau }_{xy}& {\tau }_{xz}\\ {\tau }_{xy}& {\sigma }_{yy}-{\sigma }_m& {\tau }_{yz}\\ {\tau }_{xz}& {\tau }_{yz}& {\sigma }_{zz}-{\sigma }_m\end{array}\right]}}$$ $${\sigma }_m=\frac{{\sigma }_{xx}+{\sigma }_{yy}+{\sigma }_{zz}}{3}(\text{Hydrostatic Stress})$$

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