03 Mechanical Behavior: Stress, Strain, and Deformation

Stress and strain

When a material is loaded, internal forces resist the load and the material changes shape. Stress and strain describe the relationship between applied loading and resulting deformation, helping engineers compare materials and predict whether a part will deform temporarily, deform permanently, or fail.

For a specimen pulled in tension, engineering stress is the applied force divided by the original cross-sectional area:

Here, $F$ is force and $A_0$ is original area. Stress is measured in pascals (Pa); megapascals (MPa) are common in materials engineering.

Engineering strain is the change in gauge length divided by the original gauge length:

Here, $L$ is the final gauge length, $L_0$ is the original gauge length, and $\Delta L$ is the change in length. Strain is dimensionless, though it is often expressed as a percentage. Under the usual sign convention, tensile strain is positive and compressive strain is negative.

For example, a specimen with an original area of $50\,\text{mm}^2$ carrying a tensile force of $10\,\text{kN}$ has an engineering stress of $200\,\text{MPa}$. If a gauge length of $100\,\text{mm}$ extends by $0.20\,\text{mm}$, its strain is $0.002$, or $0.2\%$.

Elastic and plastic deformation

Elastic deformation is recoverable: when the load is removed, the material returns to its original dimensions, at least approximately. In the initial linear-elastic region, stress is proportional to strain according to Hooke’s law:

The proportionality constant $E$ is Young’s modulus, or elastic modulus. It is the slope of the linear portion of a tensile stress–strain curve. A larger modulus means less elastic strain for a given stress; it does not, by itself, mean the material can withstand a greater stress before yielding.

The proportional limit is where the linear relationship ends. Some materials remain elastic beyond this point, but their stress–strain relationship is no longer linear. The elastic limit marks the approximate boundary beyond which unloading leaves a permanent change in shape. In practice, these limits can be gradual rather than sharply defined.

Plastic deformation is permanent. Once a material has yielded, removing the load does not restore its original dimensions. During unloading, the material typically recovers some elastic strain but retains permanent strain.

Reading the tensile curve

A typical engineering stress–strain curve for a ductile metal shows a sequence of behaviors and characteristic points.

1. Initial elastic region: Stress rises approximately linearly with strain. The slope gives Young’s modulus.

2. Yielding: Plastic deformation begins. Yielding may appear as a distinct point or be specified using an offset method. For many metals, the beginning of plastic deformation is reported as the yield strength.

3. Plastic deformation and strain hardening: Further deformation is permanent. In many metals, additional stress is needed to continue deforming the material; this is called strain hardening.

4. Ultimate tensile strength (UTS): This is the maximum engineering stress on the curve, calculated using the original area. In a ductile specimen, localized narrowing, or necking, generally begins around this maximum.

5. Fracture: The specimen breaks. For a specimen that necks, engineering stress may fall between the ultimate tensile strength and fracture because force is divided by the original area even as the local section continues to deform.

When there is no clear yield point, yield strength is commonly determined by the 0.2% offset method. A line parallel to the initial elastic slope is drawn from a strain of $0.002$; the intersection of that line with the curve defines the offset yield strength.

Energy absorption and fracture

The area under the curve up to a given strain represents the work done per unit original volume in a tensile test. The area up to fracture is related to the material’s toughness, its capacity to absorb energy before breaking.

A material that undergoes substantial plastic strain before fracture is called ductile. A material that fractures with little plastic deformation is called brittle.

Conditions and material comparisons

A stress–strain curve is not universal for a material under all conditions. Temperature, loading rate, processing, and specimen condition can affect measured behavior. The curve also depends on the kind of test and on whether engineering or true stress and strain are used. Engineering values use original dimensions; true values account for changing dimensions during deformation.

Together, the curve’s features allow engineers to compare stiffness, strength, ductility, and energy absorption. Stress measures applied force per area, while strain measures relative dimensional change. Young’s modulus describes the initial linear-elastic slope; elastic deformation is recoverable, whereas plastic deformation leaves a permanent change. Yield strength marks the onset of plastic behavior, ultimate tensile strength is the maximum engineering stress, and fracture marks separation of the specimen.