Product Development Engineers Ltd

Plasticity explained

Plasticity is the property of a material that allows it to undergo permanent deformation under an applied load, beyond its elastic limit, without fracture.

It is a crucial concept in solid mechanics, particularly in materials that can endure large deformations, like metals.

1. Elastic vs. Plastic Deformation

Elastic Deformation: When a material is loaded, it deforms, but once the load is removed, it returns to its original shape. This behaviour is described by Hooke’s Law: where:
- $\sigma$ is the stress,
- $\varepsilon$ is the strain,
- $E$ is the Young’s modulus (a measure of stiffness).
Plastic Deformation: Once the stress exceeds a certain point known as the yield stress ( $\sigma_y$ ), the material undergoes plastic deformation, meaning it will not return to its original shape when the load is removed.

2. Yield Criteria

To determine when a material begins to plastically deform, yield criteria are used. One common criterion is the von Mises Yield Criterion: $\sigma_{\text{eq}} = \sqrt{\frac{1}{2} \left[ (\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2 \right]} = \sigma_y$ where:

$\sigma_1, \sigma_2, \sigma_3$ are the principal stresses,
$\sigma_{\text{eq}}$ is the von Mises equivalent stress,
$\sigma_y$ is the yield stress of the material.

This criterion is widely used for ductile materials, such as metals, to predict the onset of plastic deformation.

3. Plastic Strain and Work Hardening

After yielding, as the material deforms plastically, additional stress leads to further strain. For many metals, this process also results in work hardening (also called strain hardening), which means the material becomes stronger as it is deformed. This is represented by the following relationship: $\sigma = \sigma_y + K \cdot (\varepsilon_p)^n$ where:

$K$ is the strength coefficient,
$\varepsilon_p$ is the plastic strain (total strain minus elastic strain),
$n$ is the strain hardening exponent.

In this model:

As $\varepsilon_p$ increases, $\sigma$ increases, indicating the material is hardening.

4. Plastic Strain Tensor

In three dimensions, plasticity is represented by a plastic strain tensor $\varepsilon_{ij}^p$ , which captures the components of plastic strain in different directions. The total strain tensor $\varepsilon_{ij}$ is the sum of elastic $\varepsilon_{ij}^e$ and plastic components: $\varepsilon_{ij} = \varepsilon_{ij}^e + \varepsilon_{ij}^p$ The evolution of the plastic strain tensor depends on the stress state and is often modelled using flow rules derived from plasticity theory, such as: $\dot{\varepsilon_{ij}^p} = \dot{\lambda} \frac{\partial f(\sigma_{ij})}{\partial \sigma_{ij}}$ where:

$\dot{\lambda}$ is a proportionality factor (plastic multiplier),
$f(\sigma_{ij})$ is the yield function.

5. Stress-Strain Curve

The typical stress-strain curve for a ductile material shows three regions:

Elastic Region: Follows Hooke’s Law.
Plastic Region: Begins at the yield point, with permanent deformation.
Strain Hardening Region: The material strengthens as plastic strain increases.

6. Practical Application:

In mechanical engineering, especially in structural design or metal forming, plasticity must be accounted for in material selection, failure analysis, and the design of processes that exploit plastic deformation (e.g., forging, extrusion).

October 10, 2024

PESL

Behaviours of Bodies Under Stress, Plasticity, Uncategorized

behaviours of bodies under stress, design, plasticity

Plasticity explained

1. Elastic vs. Plastic Deformation

2. Yield Criteria

3. Plastic Strain and Work Hardening

4. Plastic Strain Tensor

5. Stress-Strain Curve

6. Practical Application:

Share this:

Like this:

Leave a ReplyCancel reply

Discover more from Product Development Engineers Ltd