Thermal Stress Simulator V2

Introduction

Have you ever noticed why cork is used to seal bottles or in pin boards? When you compress cork, it does not expand much sideways, which helps it stay tightly fitted inside the bottle without slipping out. This unique behavior allows it to maintain a strong grip and perform its function effectively.

Now imagine using rubber instead. When rubber is compressed, it expands significantly in the sideways direction, which would make it lose its grip and fail as a proper seal. This difference in behavior is explained by Poisson’s ratio, which tells us how much a material expands or contracts laterally when stretched or compressed, helping us choose the right material for the right application.

What is Poisson's ratio?

Poisson's ratio is defined as the ratio of lateral strain to longitudinal strain. It is the material property and you will have different poisson's ratio for different set of material. When you apply elongation in the $x$ direction the elongation happens in $y$ and $z$ also. These elongation in $y$ and $z$ are called lateral elongation.

For this case Poisson's ratio is defined as:

$$\nu = -\frac{\text{lateral strain}}{\text{longitudinal strain}}$$

The negative sign is due to the elongation causes lateral contraction.

Interact with Poisson's ratio

To understand Poisson's ratio interactively let's start playing with this widget. First try to play with every sliders and see the changes. You can rotate the block to see more details.

• Let's make $\nu=0$, that is the Poisson's ratio for cork. Now when you apply the stress there will be only elongation and no lateral strain.
• In the second chance when you increase the $\nu=0.5$ you will notice the maximum amount of lateral strains in the prismatic bar.
• When you make the $\nu<0$ the material will expand in lateral direction with elongation and contract with longitudinal compression.

If you solve to any problem from the book you can enter the values and you can get the answers from this interactive widget.

Poisson's ratio for different materials

As discussed earlier, Poisson’s ratio is a material property, and its value varies for different materials.
Let’s look at a few common materials and their typical Poisson’s ratio values:

Material	Poisson's ratio ($\nu$)
Cork	0.0
Foam	0.10-0.50
Glass	0.18-0.3
Concrete	0.1-0.2
Steel	0.27-0.30
Copper	0.33
Rubber	0.4999

Application of Poisson's ratio

Poisson's ratio permit us to extend Hooke's law of uniaxial stress to the case of biaxial stress. Without Poisson's ratio Hooke's law for two and three dimensions were not matching with the experimental results. In 2 dimension

$$\varepsilon_x= \frac{\sigma_x}{E} -\nu\frac{\sigma_y}{E}$$ $$\varepsilon_y= \frac{\sigma_y}{E} -\nu\frac{\sigma_x}{E}$$

In these equations we have defined the strain in $y$ is affected by stress applied in $x$ direction. The strain in $x$ direction is calculated by multiplying the Poisson's ratio. In 3 dimension

$$\varepsilon_x= \frac{\sigma_x}{E} -\nu\frac{\sigma_y}{E}-\nu\frac{\sigma_z}{E}$$ $$\varepsilon_y= \frac{\sigma_y}{E} -\nu\frac{\sigma_x}{E}-\nu\frac{\sigma_z}{E}$$ $$\varepsilon_z= \frac{\sigma_z}{E} -\nu\frac{\sigma_x}{E}-\nu\frac{\sigma_y}{E}$$

What is the maximum limit of the Poisson's ratio?

To define the maximum limit of Poisson’s ratio, let us consider a three-dimensional rectangular cube subjected to stress.

$$\varepsilon_x= \frac{\sigma_x}{E} -\nu\frac{\sigma_y}{E}-\nu\frac{\sigma_z}{E}$$ $$\varepsilon_y= \frac{\sigma_y}{E} -\nu\frac{\sigma_x}{E}-\nu\frac{\sigma_z}{E}$$ $$\varepsilon_z= \frac{\sigma_z}{E} -\nu\frac{\sigma_x}{E}-\nu\frac{\sigma_y}{E}$$

Now, let's add all the strain in all three direction to get the volume dilation.

$$\varepsilon_{x}+\varepsilon_{y}+\varepsilon_{z} = \frac{1-2\nu}{E}{\sigma_x+\sigma_y+\sigma_z}$$

According to our assumptions material is homogenous and isotropic. Now let's assume that the strain is equal all three directions. That means:

$$\varepsilon_{x}+\varepsilon_{y}+\varepsilon_{z}=\varepsilon$$

Hence the equation will become:

$$\varepsilon = (1-2\nu)\frac{\sigma}{E}$$

In this equation the strain $\varepsilon$ and stress $\sigma$ must be of same sign. That means due to applied tension there must be elongation. To maintain that :

$$(1-2\nu) \ge 0$$ $$\nu = \frac{1}{2}$$

Hence the value of Poisson's ratio can not be more that 0.5. The rubber has the highest value.

Can we have negative Poisson's ratio?

Most materials have a positive Poisson’s ratio, typically ranging from 0 to 0.5. However, there are some materials—both naturally occurring and specially engineered—that exhibit a negative Poisson’s ratio.

Human skin is an example of a material that can show negative Poisson’s ratio behavior under certain conditions. Materials designed to exhibit this property are known as auxetic metamaterials.

Application of Poisson's ratio

In this section you have learned the Poisson's ratio. You have also learned the Poisson's ratio value for different materials. Let's discuss about the application of Poisson's ratio:

• Bridge cables and structural members: Tensile loading causes elongation and lateral contraction, and Poisson’s ratio helps predict these changes to prevent overstress and failure.
• Biomedical implants and prosthetics: Implants must match the longitudinal and lateral deformation of bones, where Poisson’s ratio ensures proper stress distribution and durability.
• 3D printing and additive manufacturing: Thermal and mechanical stresses during printing cause deformation governed by Poisson’s ratio, affecting warping and dimensional accuracy.
• Aerospace structures (wings and fuselage panels): Wing bending induces coupled longitudinal and lateral deformation, where Poisson’s ratio influences thickness, stiffness, and aeroelastic stability.
• Rubber seals and gaskets: High Poisson’s ratio causes rubber to expand laterally under compression, enabling effective gap filling and leak-proof sealing.