7.3 Stress-strain diagrams and Hooke's law

Stress-Strain Diagrams for Materials

Stress-strain diagrams are the primary tool for understanding how a material responds to loading. They plot the relationship between applied stress and the resulting strain, letting you identify exactly when a material will deform elastically, yield plastically, or fracture. If you can read one of these diagrams, you can predict material behavior under load and make informed design decisions.

Hooke's law governs the linear elastic portion of these diagrams. It gives you a simple proportional relationship between stress and strain, which is the basis for most structural analysis in this course.

Graphical Representation and Key Components

A stress-strain diagram is built from tensile test data. Strain (unitless, or length/length) goes on the x-axis, and stress (force/area, in Pa or psi) goes on the y-axis.

Two features of the curve carry physical meaning:

• Slope of the linear portion = the modulus of elasticity (Young's modulus, $E$). A steeper slope means a stiffer material.
• Area under the curve = energy absorbed during deformation, which relates to the material's toughness. A material with a large area under its curve can absorb more energy before fracture.

Variation in Stress-Strain Diagrams for Different Materials

The shape of a stress-strain diagram depends heavily on the type of material tested.

Ductile materials (most metals) show a clear yield point followed by a large plastic deformation region before fracture. Steel is the classic example: it has a distinct yield point, a long plastic region, and undergoes significant necking before breaking.

Brittle materials (ceramics, glass) behave very differently. Their curve is nearly linear all the way up, then the material fractures suddenly with little to no plastic deformation. Glass is a good example: it stretches elastically, then snaps without warning.

This distinction matters for design. Ductile materials give visible warning before failure (they bend and deform). Brittle materials don't.

Key Points on Stress-Strain Diagrams

Elastic and Plastic Deformation Regions

Several critical points appear on every stress-strain diagram, and you need to know what each one means:

• Proportional limit: The highest stress at which stress is directly proportional to strain. This marks the end of the straight-line portion of the curve. Beyond this point, Hooke's law no longer applies.
• Elastic limit: The maximum stress a material can sustain and still return to its original shape when unloaded. This may or may not coincide with the proportional limit. The distinction is subtle but worth noting: a material can behave nonlinearly yet still recover elastically.
• Yield point (yield strength): The stress at which permanent (plastic) deformation begins. Once you pass this point, the material won't fully return to its original dimensions. In some steels, the yield point shows up as a sharp drop in stress on the diagram, called the upper and lower yield points.

Ultimate Strength and Fracture

• Ultimate tensile strength (UTS): The highest stress value on the entire diagram. This is the maximum stress the material can withstand. On the curve, it's simply the peak.
• Fracture point: Where the material breaks completely. For ductile materials, this occurs after significant necking (localized thinning). For brittle materials, the fracture point is close to the UTS because there's almost no plastic deformation.

Some typical behaviors to keep in mind:

• Aluminum alloys have moderate ultimate strength with good ductility, so they deform noticeably before fracture.
• Concrete has relatively low tensile strength and fractures in a brittle manner (which is why it's reinforced with steel in structural applications).

Hooke's Law and Elasticity

Hooke's Law in the Elastic Region

Hooke's law states that within the proportional limit, stress is directly proportional to strain:

$\sigma = E\varepsilon$

where $\sigma$ is the applied stress, $E$ is the modulus of elasticity (Young's modulus), and $\varepsilon$ is the resulting strain.

This relationship only holds in the linear elastic region of the stress-strain diagram. Within this region, deformation is completely reversible: remove the load, and the material returns to its original shape.

Hooke's law also assumes the material is homogeneous (uniform properties throughout) and isotropic (same properties in all directions), and that deformations remain small.

Deviations from Hooke's Law

Once stress exceeds the proportional limit, the stress-strain relationship becomes nonlinear and Hooke's law no longer applies. This happens in the plastic deformation region for metals.

Some materials deviate from Hooke's law even within what might look like an "elastic" range:

• Rubber stretches elastically to very large strains, but the relationship between stress and strain is nonlinear from the start.
• Polymers can exhibit viscoelastic behavior, meaning their stress-strain response depends on how fast the load is applied and how long it's held. Time becomes a factor.

Modulus of Elasticity Calculation

Calculating the Modulus of Elasticity from Stress-Strain Data

You can determine $E$ directly from a stress-strain diagram using the slope of the linear region. Here's the process:

1. Identify the linear (straight-line) portion of the stress-strain curve.
2. Pick two points on that line: $(\varepsilon_1, \sigma_1)$ and $(\varepsilon_2, \sigma_2)$. Choose points that are well-separated for better accuracy, and make sure both fall within the proportional limit.
3. Calculate the slope:

$E = \frac{\sigma_2 - \sigma_1}{\varepsilon_2 - \varepsilon_1}$

Since strain is unitless, $E$ carries the same units as stress (Pa or psi).

Interpreting and Using the Modulus of Elasticity

The modulus of elasticity is a material property, not a structural property. It tells you how stiff a material is: how much it resists deformation per unit of applied stress.

• High $E$ = stiff material. Diamond has one of the highest moduli of elasticity (~1,220 GPa), meaning it barely deforms under load.
• Low $E$ = flexible material. Rubber has a very low modulus (~0.01–0.1 GPa), so it deforms easily.
• Steel falls in between at roughly 200 GPa, which is why it's the workhorse of structural engineering: stiff enough to limit deflection, ductile enough to warn before failure.

Once you know $E$ for a material, you can use Hooke's law to predict stress or strain for any loading condition within the elastic region. This is the foundation for most of the structural analysis you'll do in this course.