8.1 Elastic and plastic behavior of materials

Materials can behave elastically or plastically when stressed. Elastic deformation is temporary: the material returns to its original shape when the load is removed. Plastic deformation is permanent, occurring when stress exceeds the material's yield point.

Stress-strain curves capture both behaviors in a single graph, showing the transition from elastic to plastic deformation. The yield strength and ultimate tensile strength are the key parameters you'll pull from these curves, and they directly guide material selection and component design.

Elastic vs Plastic Deformation

Types of Deformation

Elastic deformation is temporary. The material returns to its original shape and size once the load is removed. In this region, the stress-strain relationship is linear, meaning stress and strain increase proportionally. Think of stretching a rubber band or compressing a spring: release the force, and the object snaps back.

Plastic deformation is permanent. The material does not return to its original shape after the load is removed. Internally, atoms have shifted to new positions, and the stress-strain relationship is no longer linear. Bending a metal paperclip past a certain point or reshaping clay are everyday examples: the new shape stays.

Yield Point and Material Behavior

The yield point is the stress at which a material transitions from elastic to plastic behavior. Below it, deformation is recoverable. Above it, permanent deformation begins. This makes the yield point one of the most critical values in engineering design.

You can find the yield point on a stress-strain curve in two ways:

1. Proportional limit method - Identify where the curve first deviates from a straight line.

2. 0.2% offset method - Draw a line parallel to the linear elastic portion of the curve, but starting at a strain of 0.002 (0.2%) on the x-axis. The stress where this line intersects the curve is the offset yield strength. This method is more common because many materials don't have a sharp, obvious yield point.

Ductile materials (steel, aluminum, copper) exhibit both elastic and plastic deformation. They can undergo significant plastic deformation before fracture, which means they can be shaped, formed, and often give visible warning before failure.

Brittle materials (glass, concrete, cast iron) show little to no plastic deformation. They fracture suddenly once their strength is exceeded, with minimal warning.

Stress-Strain Curves Analysis

Stress-Strain Curve Characteristics

A stress-strain curve plots the relationship between applied stress (y-axis) and resulting strain (x-axis) during a tensile or compressive test. The shape of this curve tells you nearly everything about how a material behaves under load.

The slope of the linear (elastic) portion of the curve is Young's modulus ($E$), also called the elastic modulus. It quantifies stiffness.

• A steeper slope means a higher $E$ and a stiffer material (diamond ≈ 1,200 GPa, steel ≈ 200 GPa).
• A shallower slope means a lower $E$ and a more flexible material (rubber ≈ 0.01–0.1 GPa, low-density polyethylene ≈ 0.2 GPa).

Key Material Properties from Stress-Strain Curves

Yield strength is the stress where plastic deformation begins. It marks the upper boundary of the elastic region. Materials with higher yield strengths (high-strength steels, titanium alloys) can carry greater loads before permanently deforming.

Ultimate tensile strength (UTS) is the maximum stress the material can withstand, corresponding to the highest point on the stress-strain curve. It represents the peak load-carrying capacity before the material begins to neck and weaken toward fracture.

Toughness is represented by the total area under the stress-strain curve. It measures the energy a material can absorb before fracturing. Tough materials (metals, composites) resist crack propagation well. Brittle materials (ceramics, glass) have small areas under their curves and absorb little energy before breaking.

Elongation at fracture (also called percent elongation) is the maximum strain the material reaches before breaking. It's a direct measure of ductility.

• High elongation: copper (~50%), gold (~45%), aluminum alloys (~10–40%)
• Low elongation: concrete (~0.01–0.02%), glass (~0%), cast iron (~0.5%)

Hooke's Law Application

Hooke's Law Formula

Hooke's Law describes the linear relationship between stress and strain in the elastic region:

$\sigma = E\epsilon$

where:

• $\sigma$ = stress (force per unit area, in Pa or MPa)
• $E$ = Young's modulus (same units as stress, Pa or GPa)
• $\epsilon$ = strain (change in length divided by original length, dimensionless)

You can rearrange this to solve for any of the three variables:

• Strain: $\epsilon = \sigma / E$
• Young's modulus: $E = \sigma / \epsilon$

Calculating Stress and Strain using Hooke's Law

Finding stress from strain:

A steel bar ($E = 200 \text{ GPa}$) experiences a strain of 0.001. What is the stress?

1. Write Hooke's Law: $\sigma = E\epsilon$
2. Substitute values: $\sigma = (200 \times 10^9 \text{ Pa})(0.001)$
3. Solve: $\sigma = 200 \times 10^6 \text{ Pa} = 200 \text{ MPa}$

Finding strain from stress:

An aluminum rod ($E = 70 \text{ GPa}$) is subjected to a stress of 100 MPa. What is the strain?

1. Rearrange: $\epsilon = \sigma / E$
2. Substitute values: $\epsilon = (100 \times 10^6 \text{ Pa}) / (70 \times 10^9 \text{ Pa})$
3. Solve: $\epsilon = 0.00143$

Validity limit: Hooke's Law only applies in the elastic region where the stress-strain curve is linear. Once you exceed the yield point and enter the plastic region, the linear relationship breaks down and permanent deformation occurs. Non-linear elastic materials like rubber or biological tissues also don't follow Hooke's Law, even in their elastic range, and require more complex models.

Yield & Tensile Strength Determination

Yield Strength Determination Methods

The yield strength defines the maximum stress a component can experience without permanently deforming. Engineers design parts to operate below this value under normal loading conditions.

Two standard methods for determining yield strength from a stress-strain curve:

1. Proportional limit method - Find the point where the curve first deviates from a straight line. This marks the end of perfectly linear-elastic behavior.

2. 0.2% offset method - Draw a line parallel to the elastic slope, offset by 0.002 strain (0.2%) along the x-axis. The intersection of this line with the stress-strain curve gives the yield strength. Most engineering references report yield strength using this method.

Several factors influence yield strength:

• Composition and processing - Alloying elements, heat treatment (quenching, tempering), and cold working can all increase yield strength in metals.
• Strain rate - Faster loading generally increases yield strength.
• Temperature - Lower temperatures tend to raise yield strength; higher temperatures lower it.

Ultimate Tensile Strength (UTS) Determination

The UTS is the highest stress value on the engineering stress-strain curve. It corresponds to the maximum load the specimen sustains, divided by the original cross-sectional area.

$\text{UTS} = \frac{F_{\text{max}}}{A_0}$

Example: A tensile specimen with a cross-sectional area of 100 mm² fails at a maximum load of 10 kN.

1. Convert units: $A_0 = 100 \text{ mm}^2 = 100 \times 10^{-6} \text{ m}^2$
2. Apply the formula: $\text{UTS} = \frac{10 \times 10^3 \text{ N}}{100 \times 10^{-6} \text{ m}^2}$
3. Solve: $\text{UTS} = 100 \times 10^6 \text{ Pa} = 100 \text{ MPa}$

For ductile materials (mild steel, aluminum alloys), there's typically a clear yield point and a large gap between yield strength and UTS. The material undergoes significant plastic deformation and necking before fracture.

For brittle materials (glass, concrete, acrylic), the yield strength and UTS are close together, or the material fractures before any measurable yielding occurs. There's little to no warning before failure.