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The problem is to predict the small strain increment () in the loaded sample, i.e. an external agency applies a small stress increment to the sample, additional to the stresses imposed by the loading apparatus.

Equivalent Spring Constant (Series)

$p=\left({\frac {\sigma _{1}'+2\sigma _{3}'}{3}}\right)$

$q={\sqrt {{\frac {1}{2}}\left(\left(\sigma _{1}'-\sigma _{2}'\right)^{2}+\left(\sigma _{1}'-\sigma _{2}'\right)^{2}+\left(\sigma _{1}'-\sigma _{2}'\right)^{2}\right)}}$

$\ {\frac {p\delta \nu }{\nu }}+q\delta \epsilon ={\frac {\delta E}{\nu }}$

$\ {\frac {p\delta \nu ^{p}}{\nu }}+q\delta \epsilon ^{p}\geq 0$

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In physics, force is any influence that causes an object with mass to change its motion or change its shape. Changing the motion of an object implies changing its linear momentum or angular momentum, or in other words, changing its velocity, i.e., to accelerate. Changing the shape of an object implies deformation of the object resulting from the redistribution of stresses. Force can also be described by intuitive concepts such as a push or a pull. A force has both magnitude and direction, making it a vector quantity. It is measured in the SI unit of newtons and represented by the symbol F.

Finding the stress components on an arbitrary plane

Figure 6. Mohr's circle for plane stress and plane strain conditions (Pole approach). Any straight line drawn from the pole will intersect the Mohr circle at a point that represents the state of stress on a plane inclined at the same orientation (parallel) in space as that line.

As mentioned before, after a stress analysis has been performed we know the stress components $\sigma _{x}$, $\sigma _{y}$, and $\tau _{xy}$ at a material point $P$. These stress components act in two perpendicular planes $A$ and $B$ passing through $P$ as shown in Figure 5 and 6. The Mohr circle is used to find the stress components $\sigma _{\mathrm {n} }$ and $\tau _{\mathrm {n} }$, i.e., coordinates of any point $D$ on the circle, acting on any other plane $D$ passing through $P$ with a direction angle $\theta$. For this, two approaches can be used: the double angle, and the Pole or origin of planes.

Double angle

As shown in Figure 5, to determine the stress components $(\sigma _{\mathrm {n} },\tau _{\mathrm {n} })$ acting on a plane $D$ at an angle $\theta$ counterclockwise to the plane $B$ on which $\sigma _{x}$ acts, we travel an angle $2\theta$ in the same counterclockwise direction around the circle from the known stress point $B(\sigma _{x},-\tau _{xy})$ to point $D(\sigma _{\mathrm {n} },\tau _{\mathrm {n} })$, i.e., an angle $2\theta$ between lines ${\overline {OB}}$ and ${\overline {OD}}$ in the Morh circle.

The double angle approach relies on the fact that the angle $\theta$ between the normal vectors to any two physical planes passing through $P$ (Figure 4) is half the angle between two lines joining their corresponding stress points $(\sigma _{\mathrm {n} },\tau _{\mathrm {n} })$ on the Morh circle and the centre of the circle.

This double angle relation comes from the fact that the parametric equations for the Mohr circle are a function of $2\theta$. It can also be seen that the planes $A$ and $B$ in the material element around $P$ of Figure 5 are separated by an angle $\theta =90^{\circ }$, which in the Mohr circle is represented by a $180^{\circ }$ angle (double the angle).

Pole or origin of planes

The second approach involves the determination of a point on the Mohr circle called the pole or the origin of planes. Any straight line drawn from the pole will intersect the Mohr circle at a point that represents the state of stress on a plane inclined at the same orientation (parallel) in space as that line. Therefore, knowing the stress components $\sigma$ and $\tau$ on any particular plane, one can draw a line parallel to that plane through the particular coordinates $\sigma _{\mathrm {n} }$ and $\tau _{\mathrm {n} }$ on the Mohr circle and find the pole as the intersection of such line with the Mohr circle. As an example, let's assume we have a state of stress with stress components $\sigma _{x},\!$, $\sigma _{y},\!$, and $\tau _{xy},\!$, as shown on Figure 6. First, we can draw a line from point $B$ parallel to the plane of action of $\sigma _{x}$, or, if we choose otherwise, a line from point $A$ parallel to the plane of action of $\sigma _{y}$. The intersection of any of these two lines with the Mohr circle is the pole. Once the pole has been determined, to find the state of stress on a plane making an angle $\theta$ with the vertical, or in other words a plane having its normal vector forming an angle $\theta$ with the horizontal plane, then we can draw a line from the pole parallel to that plane (See Figure 6). The normal and shear stresses on that plane are then the coordinates of the point of intersection between the line and the Mohr circle.

References

• Beer, Ferdinand Pierre (1992). Mechanics of Materials. McGraw-Hill Professional. ISBN 0071129391. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

• Brady, B.H.G. (1993). Rock Mechanics For Underground Mining (Third ed.). Kluwer Academic Publisher. ISBN 0412475502. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

• Dill, Ellis Harold (2006). Continuum Mechanics: Elasticity, Plasticity, Viscoelasticity. Germany: CRC Press. ISBN 0849397790.

Equations tests

$\sigma _{x}=2\mu \varepsilon _{x}+\lambda (\varepsilon _{x}+\varepsilon _{y}+\varepsilon _{z})$

$\ \tau _{xy}=\mu \gamma _{xy}$

$\ \tau _{xz}=\mu \gamma _{xz}$