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# Strain Energy Density: What, Why and How?

By Ahmad Ashwah 03 January 2022

Simulations serve as a powerful tool for engineers to predict the behaviors of their designs under certain stresses and strains. In this blog post, we’ll take a look at one of many output quantities that come from simulations in OnScale Solve: Strain Energy Density

Before we talk about strain energy density, let’s talk about strain energy as a quantity by itself.

### Strain Energy:

When a force is applied to a solid, it deforms. The work done by the applied force is stored in the solid as potential energy. Mathematically, this energy is a balance of the work done due to external loads (such as body forces, surface tractions, point loads) and volume integral of the work done due to internal stresses. The latter is referred to as strain energy.

To normalize this property throughout the material, it is easiest to think of strain energy density, which is the strain energy per unit volume. Then, total strain energy comes from the integration of strain energy density over the volume of a body. Thus, we can express strain energy density in SI units of J/m3.

Strain energy density also shares the same unit with stresses: Pascals. Another definition for strain energy can be obtained from the stress-strain diagram shown below. For a uniaxial stress state, it can be defined as the area under the curve bounded between point (0,0) where no load is applied and (εxx) at which a normal stress is applied:  In the above diagram, strain energy density is the shaded area. Once the yield strength is exceeded, permanent deformation makes these calculations more difficult, and we’ll cover them in a later blog post!

So, why is Strain Energy Density important ?

There are other material properties that can be extracted from the stress-strain curve diagram: In the elastic region, there is a maximum amount of energy that a material can absorb and still be able to “bounce back” without any permanent deformation. This is an intrinsic property and is referred to as the modulus of resilience. It is calculated as the area under the stress-strain curve bounded by (0,0) and the elastic limit (εelel) after which a material starts to yield. This is shown in the figure below. Thus, strain energy density can be used to determine if a material will yield under corresponding stress conditions. Using OnScale Solve, we can use the strain energy density output to:

1. Identify regions of highest strain energy
2. Determine how close the body is to yielding

Consider a connection rod, designed in Onshape, that is made from structural steel. A step-by-step tutorial on how to set up the problem can be found here. The design is restrained from moving in all directions at the face highlighted below. A pressure load (2 MPa) is applied at the face highlighted in the figure below: After meshing, running a simulation, and loading the results, strain energy density can be plotted. As shown below, regions of highest strain energy density can be spotted in the region colored red. However, as the maximum value of strain energy density (7.9x 103 Pa) does not exceed the modulus of resilience for structural steel (~144 x 103 Pa), it is safe to say that the structure will not yield under this load condition.

In conclusion, strain energy density is a key simulation output that can be viewed in OnScale Solve to ensure that a body part is well-engineered.

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