Mechanics of materials studies how solids deform and fail under load. Stress is force per unit area, \(\sigma = F/A\), measured in Pascals (Pa). Strain is the ratio of deformation to original length, \(\varepsilon = \Delta L / L_0\), and is dimensionless. Within the elastic limit, stress is proportional to strain according to Hooke's law, \(\sigma = E \varepsilon\), where \(E\) is Young's modulus, a measure of a material's stiffness.
Other elastic constants describe different deformation modes. Shear stress \(\tau = V/A\) acts parallel to a surface, and the shear modulus \(G = \tau / \gamma\) is the ratio of shear stress to shear strain \(\gamma\). Poisson's ratio \(\nu = -\varepsilon_{lateral}/\varepsilon_{axial}\) captures the lateral contraction that accompanies axial stretching; for most metals it lies between 0.25 and 0.35. These three constants are not independent but are related by \(G = E / [2(1+\nu)]\).
The strength of a material is described by several characteristic stresses. The yield strength is the stress at which the material begins to deform plastically (permanently). The ultimate tensile strength (UTS) is the maximum stress a material can withstand before necking begins. Engineers apply a factor of safety, \(FoS = \sigma_{failure}/\sigma_{actual}\), to ensure designs remain well below these limits. Real components can fail in other ways as well: fatigue failure occurs under repeated cyclic loading at stresses below the static UTS, while creep is the time-dependent permanent deformation of a material under constant stress at elevated temperature. Thermal expansion causes dimensions to change with temperature according to \(\Delta L = \alpha L_0 \Delta T\), where \(\alpha\) is the coefficient of thermal expansion.