Learning topic
Mechanical Properties of Materials
Stress-strain diagrams, yield strength, ultimate strength, Hooke's law, and elasticity.
Mechanical properties describe how a material deforms and fails under load. For engineering calculations, particularly important properties include elasticity, ductility, strength, and stiffness. They are determined experimentally, for example by a standard tensile test.
Stress and strain in a tensile test
The test results are represented by a stress–strain diagram in coordinates of normal stress $\sigma$ versus axial strain $\varepsilon$. Axial strain is $\varepsilon=\Delta l/l_0$ and is dimensionless.
Within the linear-elastic range, normal stress is proportional to axial strain:
$$\sigma=E\varepsilon.$$
- $\sigma$ — normal stress, Pa or MPa;
- $E$ — Young's modulus, Pa or MPa;
- $\varepsilon$ — axial strain, dimensionless.
The law applies to the linear portion of the stress–strain curve while $\sigma$ and $\varepsilon$ remain proportional.
Young's modulus $E$ characterizes material stiffness in tension and compression: a larger $E$ produces a smaller elastic strain at the same stress.
Characteristic regions of the stress–strain curve
In the initial region, stress is approximately proportional to strain. Beyond the elastic range, irreversible plastic deformation may develop. For a ductile material, commonly used characteristics include:
- proportional limit — stress up to which the $\sigma$–$\varepsilon$ relation is approximately linear;
- elastic limit — a characteristic boundary below which unloading leaves no more than a specified small permanent strain;
- yield strength — stress associated with substantial plastic deformation or defined by an offset method;
- ultimate tensile strength — the maximum engineering stress on the tensile stress–strain curve.
Elastic and plastic deformation
Elastic deformation disappears after unloading. Plastic deformation does not disappear completely, leaving a permanent change in shape or dimensions. The ability to accumulate substantial plastic deformation before fracture is called ductility.
Ductile and brittle materials
Ductile materials generally exhibit noticeable permanent deformation before fracture. Brittle materials fracture with relatively little plastic deformation. This distinction influences the choice of design strength and safety factor.
Strain hardening
Plastic deformation can change material properties. During cold plastic deformation, strain hardening commonly increases resistance to further plastic flow while reducing the remaining ductility.
Allowable stress and factor of safety
In allowable-stress design, a limiting material strength is reduced by a factor of safety:
$$[\sigma]=\frac{\sigma_{\mathrm{lim}}}{n}.$$
- [σ] — allowable normal stress;
- σlim — selected limiting material strength;
- n — factor of safety, n > 1.
The choice of σlim depends on the material and design method: ductile materials are often referenced to yield strength, while brittle materials may be referenced to a fracture strength. The strength condition is $|\sigma_{\max}|\le[\sigma]$.
The factor of safety accounts for uncertainty in loads, material properties, the calculation model, and operating conditions. Its value is not universal and must follow the adopted design method or code.
Example
Suppose $E=200\ \text{GPa}$ and the elastic strain is $\varepsilon=0.001$. Hooke's law gives:
$$\sigma=E\varepsilon=200\cdot10^9\cdot0.001=200\cdot10^6\ \text{Pa}=200\ \text{MPa}.$$
If the selected limiting material strength is $360\ \text{MPa}$ and $n=1.5$, the allowable stress is $[\sigma]=360/1.5=240\ \text{MPa}$. Thus $200\ \text{MPa}$ does not exceed $240\ \text{MPa}$.
Learning outcome
After studying this topic, you should be able to distinguish elastic and plastic deformation, interpret the main regions of a tensile stress–strain curve, explain Young's modulus and strength characteristics, apply Hooke's law within its valid range, and perform a basic allowable-stress check.
About this topic
Experimental testing of mechanical properties forms the foundation for proper material selection in engineering. This section analyzes the standard tensile stress-strain diagram for mild steel, highlighting proportionality, elasticity, yield, and ultimate strength limits. Differences between ductile and brittle materials, plastic deformation, strain hardening, and safety factors are thoroughly explained alongside allowable stress calculations.