1.4. Mohr’s circle

We can also think of Mohr’s circle for strain. Mohr’s circle for stress gave the value of normal traction (σ) and shear traction (τ) on an arbitrary plane. Similarly, if we know the value of longitudinal strain along two perpendicular directions say ϵ and ϵ and also know shear strain between ϵ and ϵ, then one can use Mohr’s circle for strain to obtain longitudinal and shear strain for two perpendicular line elements which are oriented at angle θ relative to ϵ and ϵ pairs.

Mohr's Circle is a graphical representation that helps in understanding the relationship between normal and shear strains. Just like it was used for stress, we can adapt its principles for strain. When we have longitudinal strain values along two perpendicular directions as well as shear strain, we can use these to construct Mohr's Circle for strain, allowing us to determine the effects of the strains on line elements at an angle with respect to those principal directions.

Imagine a rubber band that can stretch in two directions. If you know how much it stretches in each direction (the longitudinal strain) and the additional twist or shear strain between those two stretches, you can visualize how the rubber band will appear when you look at it at an angle. Mohr's Circle provides a clear way to visualize and calculate the resulting strains from these defined conditions.

The strain plane will have its axes as longitudinal strain (ϵ) and shear strain (γ). However, as the formula for shear strain has an extra factor of 2 when compared with the formula for shear stress, we need to keep the vertical axis in the strain plane as γ/2. This will permit us to draw Mohr’s Circle for strain in exactly the same manner as we draw Mohr’s Circle for stress.

In Mohr's Circle for stresses, shear stress is represented directly. For strains, however, the representation requires an adjustment because of the definition of shear strain, which effectively doubles the value due to the geometrical relationships involved. Therefore, we use γ/2 on the vertical axis, allowing the diagram to maintain its relevance and function effectively.

Think of trying to visualize a seesaw. If one side of the seesaw represents one type of strain, and the other side represents another, the amount they tip relative to each other is akin to you needing to adjust how you measure their comfort level during play. By changing how you record their positions, you can still get a good sense of how they interact, just like using the γ/2 helps in understanding the strain relationships.

For drawing the Mohr’s circle, we need to find out the center and the radius of the circle. We first draw the point corresponding to line elements along x and y directions. Thus, we mark the point (ϵ_xx, ϵ_yy) as shown in Figure 3. The center of the circle will be at the midpoint of ϵ_xx and ϵ_yy on the axis. Now, we have the center of the circle and a point on the circle (ϵ_xx, ϵ_yy). Thus, we can get the radius by joining them as shown in Figure 3. We can then draw the circle itself with the center and radius known.

To effectively use Mohr's Circle, we establish the center as the mid-point of our longitudinal strain values along the axes. By marking these points and determining the distance or radius from the center to the points along the strain axes, we can accurately sketch the circle, making it possible to visualize various strain states and their interactions.

Imagine using a compass to draw a circle; the needle is your center point, while the pencil is the point you’ll use to create the circle. The distance from the needle to the pencil is the radius. By knowing where to place your needle and how far to stretch your pencil, you can draw the perfect circle – much like how we use our strain values to determine points on the Mohr's Circle.

We can extract a lot of information from this circle just like we had seen in the stress case. For example, the principal strain components will be obtained from the points where the circle cuts the axis (shown in red crossed circles in Figure 3). Thus, we have Principal strain components: ϵ_1, ϵ_2.

The intersections of Mohr's Circle with the axes correspond to the maximum and minimum principal strains. By identifying these points where the circle meets the axes, we can determine the principal strain values, allowing us to analyze the deformation behavior of the material thoroughly.

Think of a target board used in archery – where the arrows hit the bullseye represents key points of interest, just like the circumferences of Mohr's Circle represent critical strain points on our material. The focus is on accuracy and the corresponding values shared with the outer parts of the board, reminding us of how precision plays a role in understanding the strain distributions in our material.

Also, maximum shear strain (γ_max) can be obtained from the Mohr’s circle. γ_max is equal to the radius of the Mohr’s circle. Thus, γ_max = 2R.

From the drawing of Mohr's Circle, the maximum shear strain can be deduced as the radius of the circle. By using the relationship that the maximum shear strain is twice the radius, we can derive important strain information. This is vital in structural contexts where shear failure may occur.

Picture a balloon being inflated. The maximum stretch (or shear strain) it can handle before bursting correlates to how far you can push the limits. Similarly, determining the maximum shear strain through Mohr's Circle gives you an understanding of how much change or stress the material can safely endure.

Finally, we can find longitudinal and shear strains for any arbitrary line elements. For example, to get the longitudinal strain for a line element which makes an angle θ with the e_1 direction in the clockwise direction, we need to go anticlockwise by 2θ on the Mohr’s circle from (ϵ_xx). For getting the shear strain, we should remember that we need to multiply the value obtained from Mohr’s circle graph by 2.

This step allows for versatility in applying Mohr's Circle to various angles of application. By moving counter-clockwise by double the angle on the circle, we can account for the differing orientations of line elements, facilitating calculations for a variety of practical applications.

It's akin to rotating a map to measure a distance to the next town. Depending on how you turn your map, the distance to travel changes. Similarly, adjusting our angle in the Mohr's Circle allows us to accommodate the effects of direction changes on strain, helping us to pinpoint exactly how a material responds under different conditions.

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