# Elastic and Plastic Deformation ## Understanding Deformation Types - Materials undergo two distinct types of deformation when forces are applied to them: - **Elastic deformation** - **Plastic deformation** ### Elastic Deformation - **Elastic deformation** occurs when a material returns to its original shape after the applied force is removed - The object behaves like a spring - bouncing back to its starting configuration - This type of deformation is **reversible** - Example: stretching a rubber band within reasonable limits, then releasing it - On a force-extension graph, elastic deformation is shown in the **elastic region** - This is the straight-line portion of the graph before the elastic limit - The material obeys Hooke's Law in this region

Elastic deformation: when the load is removed, the object returns to its original shape

### Plastic Deformation - **Plastic deformation** occurs when a material is permanently changed - The material does **not** return to its original shape or length when the force is removed - This type of deformation is **permanent** - Example: bending a paperclip repeatedly until it stays bent - Plastic deformation occurs **beyond the elastic limit** - This is shown in the plastic region of force-extension graphs - The graph typically curves in this region - Extension is no longer proportional to the applied force

Beyond the elastic limit, materials experience permanent plastic deformation

## The Elastic Limit - The **elastic limit** is a critical point on force-extension graphs - It divides the elastic region from the plastic region - Before this point: material behaves elastically - After this point: material behaves plastically - The elastic limit represents the maximum force that can be applied before permanent deformation occurs ## Identifying Regions on Force-Extension Graphs ### Elastic Region - Characteristics: - Extension is **directly proportional** to the applied force - Appears as a **straight line** on the graph - Material obeys Hooke's Law: $F = kx$ - Starts at the origin and extends to the elastic limit ### Plastic Region - Characteristics: - Extension is **no longer proportional** to the applied force - The graph begins to **curve** (typically flattening out) - Starts at the elastic limit - Ends at the breaking point (fracture) | Region | Graph Appearance | Material Behaviour | After Force Removal | |--------|------------------|-------------------|---------------------| | Elastic | Straight line | Obeys Hooke's Law | Returns to original shape | | Plastic | Curved line | Does not obey Hooke's Law | Permanently deformed | ### Typical Force-Extension Behaviour The graph below shows a typical ductile material's behaviour under tensile force. Notice the distinct transition from the linear elastic region to the curved plastic region:

## Material Properties: Brittle vs Ductile ### Brittle Materials

Brittle materials fracture before plastic deformation, showing little to no plastic behaviour

- **Brittle materials** break suddenly with minimal warning - Very little or **no plastic region** on their force-extension graph - The material snaps with insignificant plastic deformation - Examples: glass, ceramic, concrete, cast iron - On a stress-strain or force-extension graph: - Represented by a **straight line** through the origin - Minimal or negligible curved region before fracture - Sudden breakpoint - Properties: - High breaking stress - Low strain before breaking - Cannot be easily reshaped ### Ductile Materials - **Ductile materials** can withstand large plastic deformation without breaking - Have a **larger plastic region** on their force-extension graph - The material stretches into a new shape before eventually breaking - Examples: copper, aluminium, gold, mild steel - On a stress-strain or force-extension graph: - Straight line through the origin (elastic region) - Then curves significantly towards the x-axis (plastic region) - Extended curved region before breaking - Properties: - Can be drawn into wires - Can be hammered into thin sheets - Undergo substantial extension before fracture

Ductile materials are useful in engineering because they deform visibly before breaking, providing warning of failure

### Comparing Brittle and Ductile Behaviour The graph below contrasts the force-extension behaviour of a brittle material (e.g., glass) versus a ductile material (e.g., copper):

## Loading and Unloading Curves ### Within the Elastic Limit - If a material is stretched within its elastic limit: - The **loading curve** (force applied) and **unloading curve** (force removed) are the **same** - The material returns along the same path on the graph - No permanent extension remains ### Beyond the Elastic Limit - If a material is stretched beyond its elastic limit into the plastic region: - Plastic deformation has occurred - When the force is removed, the material does **not** return to the origin - The **unloading curve** has the same gradient as the loading curve - The unloading curve intercepts the x-axis at a positive value (permanent extension) - The y-intercept is no longer at the origin because: - Bonds in the material have been re-aligned during plastic deformation - Some permanent extension remains even with zero applied force - The gradient remains the same because: - The intermolecular forces (forces between bonds) are identical to before - The spring constant $k$ of the material hasn't changed ### Visualizing Loading and Unloading The graphs below show the loading phase (force increasing) and unloading phase (force decreasing) when a material is stretched beyond its elastic limit. Note the permanent extension that remains: **Loading Curve:**

**Unloading Curve:**

Notice that when the force returns to zero, the extension remains at 3.5 mm (permanent extension), not at the origin. ## Practical Examples ### Example: Metal Wire Behaviour Consider a copper wire subjected to increasing tensile force: 1. **Elastic Region**: - Wire extends proportionally - Removing the force returns wire to original length - Bonds between atoms stretch but don't break 2. **At Elastic Limit**: - Maximum stress before permanent deformation - Critical transition point 3. **Plastic Region**: - Wire continues to extend with less additional force - Atomic planes slide past each other - Removing force leaves permanent extension 4. **Breaking Point**: - Wire fractures - Ultimate tensile stress exceeded ### Comparing Material Responses **Scenario**: Three identical masses are hung from samples of different materials - **Glass (brittle)**: - Extends slightly - Snaps suddenly with little warning - No plastic deformation - **Copper (ductile)**: - Extends elastically at first - Begins to "neck" (narrow) as plastic deformation occurs - Stretches significantly before breaking - **Rubber (polymeric)**: - Large elastic extension - Returns to nearly original shape (with some energy loss as heat) - Different behaviour entirely (covered in stress-strain graphs) ## Key Points to Remember - Elastic deformation is **reversible**; plastic deformation is **permanent** - The elastic limit separates these two regions - Brittle materials show minimal plastic behaviour before fracturing - Ductile materials show extensive plastic behaviour before fracturing - The gradient of the unloading curve equals the gradient of the loading curve - Permanent extension results when materials are stretched beyond their elastic limit