Simulate Early, Simulate Often... In Rhino

So, how are stress and strain connected? For isotropic materials, such as most metals, there are two values that relate values of stress and strain and fully describe a material's behavior when forces are applied. These are the **elastic modulus** and **Poisson's ratio**.

The elastic modulus is a measure of the stiffness of a material. For a given value of stress pulling on an object, increasing the elastic modulus (increasing the stiffness) causes less strain (less deformation of the object). The following Scan&Solve model shows the effect of changing the elastic modulus.

Try it here! Run SnS on both cylinders: ElasticModulusCylinders.3dm

The colors show the stress in the bars. However, the stress applied at the ends is the same. Instead, the bars differ in material. The top bar is made of steel with an elastic modulus of 200 MPa, while the middle bar is made of brass with an elastic modulus of 100 MPa. The top bar is then expected to be twice as stiff as the middle bar. This is seen in their deformation: the top bar strains half much as the middle bar.

Another way a material can deform is through the Poisson effect. Whenever a material is strained in one direction the material will shrink or expand in orthogonal directions. The Poisson effect can be observed by stretching a rubber band. The rubber will extend in the direction it is pulled, but will also get thinner and narrower.

Run SnS and look at the Y-direction deformation: PoissonBeam.3dm

In the example above, the object is being pulled along the X axis. It is also contracting in the orthogonal directions because of the Poisson effect. The colors show the Y direction displacement (green line), with red indicating displacement in the positive Y-direction and blue indicating a displacement in the negative Y-direction. As expected, the object is getting thinner and narrower as it is pulled.

The Poisson ratio describes the behavior of the Poisson effect. For a given stress pulling on an object, a small Poisson ratio will cause less contraction while a large Poisson ratio indicates the material will contract significantly. The effect of Poisson’s ratio on the contraction in the orthogonal directions is shown below. Both blocks have an identical stress applied in the vertical direction.

Try this simulation with the custom materials provided. Poisson1 is the low ratio material and Poisson2

is the high ratio material: Cork&Rubber.3dm

The colors indicate strain in the vertical direction. Note that the strain is the same because the blocks and the stress applied are the same. The only difference is in their Poisson ratio. The block on the left has a very small Poisson ratio (like cork), and does not contract much in the horizontal direction. The block on the right has a large Poisson ratio (like rubber), and contracts significantly in the horizontal direction. All normal materials have a Poisson ratio between 0 and 0.5. These extremes are shown above: cork's Poisson ratio is almost 0 and rubber's Poisson ratio is near 0.5.

The expansion or contraction due to the Poisson effect also causes stress within a material. This stress can be significant near restrained areas of models. In the following example, the blue area of the block has been restrained. The block is being pushed on its end, and is contracting in the same direction while expanding in the orthogonal directions.

Run SnS on this model and see the large stress at the restraint boundary: PoissonRestrained.3dm

The red area indicates high stress levels within the material. This is caused by the sharp change in deformation between the restrained and unrestrained region. Near the applied stress at the end, the material has expanded due to the Poisson effect. However, the material must contract significantly at the boundary between the restrained and unrestrained part of the block, causing the high stress level shown.

As long as the material is not yielding, the elastic modulus and Poisson ratio are all that is needed to analyze material behavior under load. For Scan&Solve's default materials, the elastic modulus and Poisson ratio can be seen in the custom material interface. These two material parameters play an important role in material selection for designs, particularly when the deflections are to be minimized.

Thanks again for reading,

Will

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