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# Simulating plastic materials in OnScale Solve

By Tristan Akalumhe 30 January 2023

What is plastic material behavior and why is it important? In this blog post, we will introduce plasticity and show how it can be simulated in OnScale Solve.

## What are Plastic material models?

First, it is essential to have at least a basic understanding of the concepts of stress and strain in an object. When an external force is applied to an object, the stress in the object is a measure of the amount of load that is carried at any given point in the material that makes up the object. Furthermore, the strain is a measure of how much the material deforms at that point. If the stress is plotted versus the strain, the resulting stress-strain curve of the material provides insight about how the material deforms when subjected to a load. A typical stress-strain curve for a metal is presented in fig. 1.

Without going into details, materials are typically characterized using laboratory tests like uniaxial tension tests to create stress-strain curves in terms of a scalar equivalent stress measure like the von Mises stress and an equivalent strain.

The first stage, where the stress is less than the yield strength, is known as the linear elastic region. This stage is governed by Hooke’s law which states that the stress \sigma is linearly proportional to the strain \epsilon.

\sigma = E \cdot \epsilon

Here E is the proportionality constant known as the Young’s Modulus of the material. At this stage, a part or specimen can return to its original dimension when unloaded, hence its elastic behavior. The end of this stage is the initiation point for plastic material deformation and the stress magnitude at this point is equal to the yield strength.

As additional load is applied to a material that has reached its yield point, the material deforms more for a given increment of load than it did in the elastic stage, shown by the slope of the stress-strain curve after the yield point (the tangent modulus) being less than the slope of the curve in the elastic stage. When the tangent modulus is positive, it indicates that the yield strength is increasing so this portion of the curve is often called the hardening curve.

Deformation in the material after it yields includes, a permanent part that is not recovered when the load is removed - this is the plastic deformation. A small amount of elastic strain is recovered on unloading and if the material is reloaded, it will deform elastically up until the stress reaches a new yield strength. If you are interested, more detailed theory on plastic materials and other nonlinearities was presented in this blogpost.

## Why are Plastic material models important in FEA simulation software?

If the stress in an object is kept below the yield strength, the material can be expected to behave elastically without permanent deformation. Thus, structures, parts, and assemblies are typically designed to function in this elastic region. In cases where a material may be loaded beyond it’s yield strength it is usually important to know how much deformation the object will undergo. In some cases, plastic deformation may actually be part of the design. These scenarios can be investigated by including plastic behavior in the material property definitions in OnScale Solve.

It is important to be aware of the yield strength of materials in objects being simulated. The stress results of a simulation using only linear-elastic material properties should be analyzed to confirm that the stress at all material points is less then the yield strength. fig. 2 schematically shows the difference in predicted stress values in material loaded beyond the yield strength in a linear-elastic simulation (represented by the dashed line) and the correct stress value (the solid line) obtained from a nonlinear plastic simulation.

If a simulation is run with only linear elastic material properties and the stress results are close to or greater than the yield strength, the simulation can be run again with plastic behavior activated to investigate the extent of plastic deformation. Results in OnScale Solve include the full plastic strain tensor and the equivalent plastic strain when plastic materials are included in the simulation.

## Let’s Dive into an Example in OnScale Solve

The following analysis was simulated with small deformation assumptions. That is, the material is expected to yield, but overall strains will be small, which we will verify. A following blog post will discuss simulations where large deformation analysis is required.

Table 1: Values of Material properties.
Mechanical properties (symbol)Value (unit)
Young’s modulus (E)205~\text{GPa}
Poisson’s ratio (\nu)0.3
Yield strength (\sigma_y)250~\text{MPa}
Isotropic hardening modulus (H)0~\text{GPa}

We will analyze the welded I-beam joint in fig. 3 with mechanical material properties presented in tbl. 1. The steel material is represented with perfectly plastic hardening beyond its yield point, as indicated by its isotropic hardening modulus of 0. This is often a valid assumption for structural steel alloys undergoing only small strains because of the perfectly plastic plateau exhibited by these materials immediately after the yield point.

The I-beam model is available as an Onshape document for download. I-Beams are widely used to support structural loads in buildings, bridges and other infrastructure. This problem is set up to analyze the deformation in a welded joint due to dead loads on the supporting beams from the building weight.

The sectioned faces of the column are fully restrained in this problem. Given that the beams in this sectioned model are cut short, we can reproduce a real loading condition with a shear force and a torque applied to the sectioned beam faces, as shown in fig. 3 and fig. 4 respectively. In this example, the connection beam end plates are fully bonded to the column flanges to represent a perfect weld. Simulation results are presented in tbl. 2.

Table 2: Simulation results.
Simulation outputLinear-elastic ModelPlastic Model
Maximum Displacement10.14~\text{mm}14.03~\text{mm}
Maximum Equivalent Plastic strain0.001.08e-2

Including the plastic material model allows us to see the permanent deformation around the joint of the beam to the column. For example, the elastic model gives us the impression that there is no plastic strain in the model around the weld while the plastic model reports a small amount plastic strain when the material yields. In addition, a significantly higher displacement is reported with the plastic model. Therefore, Design engineers are well equiped to make the right decision with plastic material models in FEA software.

In conclusion, we briefly discussed what plastic materials are, its importance in FEA simulation software, and covered a practical example of its usage in OnScale Solve.