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# Young’s Modulus

Young’s modulus or elastic modulus is a mechanical property of linear elastic solid materials. It defines the relationship between stress and strain in a solid material, and is given by:

$$\sigma = E \varepsilon$$

where,

• $$E$$ is Young’s modulus of the material,
• $$\sigma$$ is the stress, and
• $$\varepsilon$$ is the strain.

Young’s modulus has units of pressure or stress ($$Pa, psi$$) since the strain is a dimensionless quantity. From the given relation, it can be seen that the elastic modulus is the ratio between stress and strain:

$$E = \frac{ \sigma }{ \varepsilon }$$

Physically speaking, it is a measure of how rigid a material is. As the overall rigidity of a part is also dependent on the geometrical shape, it is necessary to have a shape-independent property to measure the material.

## Young’s Modulus in Linear Elasticity

A solid material is called elastic if it is able to naturally return to its original shape after the applied loads are removed. On the other hand, if the ratio between stress and strain is constant during the deformation process, then Young’s modulus is also a constant, and the material is considered to be linear elastic.

## Young’s Modulus in Nonlinear Materials

In general, all solid materials are not linear elastic, especially when they are subjected to high levels of deformation. Figure 2 shows an example stress-strain curve for a ductile carbon steel material, as obtained from a uniaxial tension test: Figure 2: Real stress-strain curve for a carbon steel material. The material has a linear elastic behavior only for small strain values.

It can be seen that the material has a linear elastic behavior only for small strain values. When deformation increases, nonlinear behavior starts to occur. In this case, the ductile material exhibits plasticity, but other nonlinearities such as hyperelasticity or fragile rupture can also occur.

In general, most materials can be approximated to be linear elastic under the assumption of small deformation. In that case, the elastic modulus can be approximated by analyzing the linear portion of the stress-strain curve, and computing:

$$E_y = \frac{ \sigma_y }{ \epsilon_y }$$

where,

• $$E_y$$ is the linear elastic Young’s modulus approximation,
• $$\sigma_y$$ is the stress value at the elastic limit (end of the straight line), and
• $$\varepsilon_y$$ is the strain value at the elastic limit.

## Workbench Setup

When setting up a structural simulation, Young’s modulus parameter for a material must be specified in the case of two models: linear elastic and plastic materials. Figure 3: Setting up (E) Young’s modulus inside the material parameters panel for linear elastic and plastic materials

Linear Elastic vs. Plastic

• Linear elastic: The modulus determines the proportional relation between stress and strain, as described above.
• Plastic: The modulus determines the linear portion of the deformation curve, where the stress-strain state is below the yield point. After the material crosses this threshold, the plastic deformation takes control.
• Last updated: November 10th, 2020