4 Mechanical Behavior of Materials

Stress and strain

Materials respond to applied loads by changing shape or size. Understanding that response helps engineers predict whether a component will deform reversibly, deform permanently, or fail.

For a straight specimen under axial tension or compression, average engineering stress is calculated using the applied force and the specimen’s original cross-sectional area:

Here, $F$ is the applied force and $A_0$ is the original cross-sectional area. Stress is measured in pascals (Pa), commonly megapascals (MPa). Engineering stress uses the original area. After substantial deformation, especially necking, the actual area becomes smaller, so engineering stress can differ from stress calculated using the current area.

Strain measures deformation relative to the original size. For axial loading:

In this expression, $L_0$ is initial length, $L$ is loaded length, and $\Delta L$ is the change in length. Strain has no units. Under the usual sign convention, tensile strain is positive when a specimen elongates, while compressive strain is negative.

Elastic and plastic behavior

A tensile test records stress as a specimen is stretched, producing a Stress–strain curve. Its initial slope indicates elastic modulus; the yield region indicates the onset of permanent deformation; its peak gives ultimate tensile strength; and its endpoint records fracture.

Elastic deformation is recoverable: after the load is removed, the material returns to its original shape approximately or completely. In the initial, nearly linear portion of many curves, stress is proportional to strain:

Here, $E$ is Young’s modulus, the slope of the linear elastic curve. It measures a material’s resistance to elastic deformation. A high modulus means a material strains less under a given stress, but does not by itself mean the material has high strength. Some materials remain elastic beyond the linear region, where stress and strain are no longer proportional.

Plastic deformation is permanent. If a specimen is loaded beyond its elastic limit and then unloaded, it retains some residual strain. On a tensile curve, yield strength marks the onset of significant permanent deformation. For materials without a sharp yield point, it is commonly reported using a specified offset method, such as the $0.2\%$ offset.

Stiffness, strength, ductility, and hardness

Stiffness, strength, ductility, and hardness describe different aspects of mechanical behavior; they are not interchangeable measures.

• Stiffness is resistance to deformation. Young’s modulus describes material stiffness, while the stiffness of a particular component also depends on its geometry. For example, under the same force, a longer or thinner rod stretches more than a shorter or thicker rod made from the same material.

• Strength is the stress a material can withstand under specified conditions. Yield strength indicates resistance to permanent deformation, while ultimate tensile strength is the maximum engineering stress reached in a tensile test. Neither value alone specifies resistance to every type of failure or loading.

• Ductility is the extent of plastic deformation a material can undergo before fracture. It is often measured by elongation or reduction in area after a tensile test. A ductile metal can stretch substantially before breaking; many ceramics fracture with little plastic deformation.

• Hardness is resistance to localized deformation, commonly measured by pressing an indenter into a surface or by a scratch test. Brinell, Vickers, and Rockwell are common indentation methods. Hardness is useful for comparison and quality control, but is not interchangeable with strength, toughness, or ductility.

The area beneath a stress–strain curve up to fracture represents energy absorbed per unit volume. A material that absorbs substantial energy before breaking is described as tough. Hardness alone does not establish toughness: a hard material may resist indentation but still fracture suddenly.

Time and temperature effects

Some materials do not have a single stress–strain response independent of time. Creep is the gradual increase in strain while stress is held approximately constant. In many crystalline materials, creep curves can include an initial period of decreasing creep rate, a more nearly steady-rate stage, and a final stage of accelerating deformation before rupture. Creep generally becomes more important as temperature or applied stress increases.

Many polymers are viscoelastic: their response combines elastic recovery with time-dependent deformation. Under a constant load, a polymer may continue to stretch through creep. Under a fixed deformation, the stress it sustains may decrease through stress relaxation. Both responses depend on temperature and loading timescale. For instance, a polymer component supporting a steady load may sag more over time, particularly at higher temperature.

Temperature can also change modulus, strength, and ductility. Mechanical properties should therefore be measured or selected for the relevant temperature, loading rate, and service duration, rather than assumed to be universal constants. A room-temperature, short-duration tensile test may not predict long-term behavior at elevated temperature.

Using curves to anticipate failure

A ductile specimen may undergo substantial plastic strain before breaking, while a brittle specimen may fracture soon after the elastic range. When designing a component, allowable stress must account for the relevant failure mode and operating conditions; modulus or hardness alone is not enough to establish an appropriate limit.

Stress measures force per area, while strain measures relative deformation. Elastic deformation is recoverable and plastic deformation remains after unloading. Young’s modulus measures elastic stiffness, whereas yield and ultimate strengths describe different limits on load-bearing behavior. Ductility measures plastic deformation before fracture, and hardness measures resistance to localized surface deformation. Creep, stress relaxation, and other time- or temperature-dependent effects can substantially change performance in service.