04 Mechanical Properties of Materials

Stress, strain, and stiffness

Mechanical properties describe how a material responds to forces, deformation, time, and failure. They help engineers judge whether a material can carry a load safely, resist wear, or operate for a required service life.

Stress and strain

Stress measures internal force per unit area. For a simple axial load, engineering stress is

where $F$ is the applied force and $A_0$ is the specimen’s original cross-sectional area. Stress is measured in pascals, commonly megapascals in engineering. Tensile stress pulls a material apart, compressive stress pushes it together, and shear stress acts parallel to a surface.

Strain measures deformation relative to an original dimension. Engineering tensile strain is

where $L_0$ is the original length and $L$ is the length under load. Strain has no units. For example, a bar initially $1\,\mathrm{m}$ long that elongates by $1\,\mathrm{mm}$ has a strain of $0.001$, or $0.1\%$.

Stiffness and strength

A tensile test records stress as a function of strain. In the initial, approximately straight portion of the curve, many materials follow Hooke’s law:

where $E$ is Young’s modulus, a measure of stiffness. A stiff material deforms relatively little under a given elastic stress. Stiffness is distinct from strength: a material can be stiff but fail at a comparatively low stress.

Elastic and plastic deformation

Elastic deformation is recoverable: when the load is removed, the material returns to its prior dimensions. The stress–strain relation is often linear at small strains, but elastic behavior can also be nonlinear.

Yield strength marks the onset of significant permanent deformation. If a clear yield point is absent, it is commonly determined using the $0.2\%$ offset method. Beyond yielding, plastic deformation leaves a permanent change in shape.

In ductile metals, plastic deformation is largely associated with dislocations moving through the crystal structure. As deformation continues, many metals become harder to deform; this is strain hardening. In a tensile test, a specimen usually reaches its maximum engineering stress, called ultimate tensile strength (UTS), before localized thinning, or necking, develops. The specimen eventually fractures. Ductility is often reported as elongation or reduction in area at fracture.

Strengthening mechanisms

Strengthening usually means making plastic deformation more difficult, particularly by impeding dislocation motion. Common methods include:

• Grain refinement: Grain boundaries obstruct dislocations. Smaller grains often increase yield strength; the Hall–Petch relation expresses this trend over many ranges of grain size.

• Solid-solution strengthening: Solute atoms distort the crystal lattice and hinder dislocation movement, so alloying can strengthen a metal.

• Precipitation strengthening: Fine particles within a metal obstruct dislocations. Heat treatment can control the size and distribution of these precipitates.

• Strain hardening: Plastic working increases dislocation density, making further dislocation motion more difficult. This can increase strength while reducing ductility.

• Transformation strengthening: In some alloys, heat treatment creates a harder microstructure that resists deformation.

Strengthening involves trade-offs. A stronger condition may be less ductile or tougher, and processing can affect other properties as well.

Fracture and crack growth

Fracture is the separation of a material into parts. A ductile material may undergo substantial plastic deformation before breaking, whereas a brittle material may fracture with little visible deformation. Actual behavior depends on factors including temperature, loading rate, microstructure, and flaws.

A crack or sharp defect concentrates stress near its tip. Fracture toughness, often denoted $K_{IC}$ for plane-strain mode-I toughness, describes resistance to crack growth under specified conditions. In linear-elastic fracture mechanics, a crack becomes unstable when the stress intensity reaches the material’s fracture toughness:

where $\sigma$ is the applied stress, $a$ is a characteristic crack size, and $Y$ is a factor determined by geometry and loading. A component can therefore fracture below its nominal strength if a sufficiently large flaw is present. Inspection, flaw control, and appropriate design margins help reduce this risk.

Fatigue under cyclic loading

Fatigue is damage caused by repeated or fluctuating loading. A component can fail after many cycles even when the stress is below the yield strength measured in a single loading event. Fatigue cracks often begin at surface defects, notches, or other stress concentrations, then grow with continued cycling until the remaining section fractures.

An S–N curve plots cyclic stress amplitude against cycles to failure. For some steels, a fatigue limit is observed under particular test conditions; many materials instead show decreasing fatigue strength as the required number of cycles increases. Fatigue life depends on stress range, mean stress, geometry, surface condition, environment, and material condition. Rounded transitions, smooth surfaces, and avoiding unnecessary cyclic loads can improve fatigue performance.

Creep over time

Creep is time-dependent deformation under sustained stress. It becomes especially important at elevated temperatures, though some polymers and other materials can creep appreciably at ordinary service temperatures.

A typical creep curve has three stages:

1. Primary creep: The creep rate decreases with time.

2. Secondary creep: Deformation proceeds at a nearly steady rate.

3. Tertiary creep: The rate accelerates, often as damage accumulates, leading to rupture.

Creep assessment must consider both temperature and duration, not just the initial applied load. Engineers use time-dependent deformation and rupture data to set operating limits for components such as hot pipes, turbine parts, and furnace equipment.

Hardness and its limits

Hardness is resistance to localized indentation or scratching. In common indentation tests, an indenter is pressed into the surface under a specified load, and the resulting impression is measured. Brinell, Rockwell, and Vickers tests use different indenters and procedures, so hardness values should be reported with the test method and conditions.

Hardness can provide a quick quality-control measure and often correlates with strength within a particular material family and processing condition. It is not a universal substitute for tensile strength, toughness, or fatigue testing: materials with similar hardness may behave differently under other kinds of loading.