Mechanical behaviour of materials

Explain how a tensile test links stress and strain in elastic deformation, and how the slope gives Young's modulus, with Poisson's ratio and Hooke's law in uniaxial tension.

Explore elastic constants—Hooke's law in one dimension, Young's modulus, Poisson's ratio, shear modulus, and bulk modulus—derive isotropic relations, reveal Lamb's constant, and examine elastic energy and deviatoric versus hydrostatic stresses.

Explore how elastic constants reveal material behavior by linking force, displacement, and strain through a potential-energy curve; curvature defines the elastic modulus and depends on orientation and temperature.

Explore generalized Hooke's law for anisotropic materials, using the 4th-order stiffness tensor and Voigt notation, and count elastic constants for isotropic, orthotropic, cubic, tetragonal, hexagonal, orthorhombic, monoclinic, and trigonal systems.

Analyze the stress-strain curve from linear elastic to plastic deformation, covering the 0.2% offset yield strength, ultimate tensile strength, ductility, toughness, and true stress and true strain.

Explains elastic behavior with Hooke's law up to yield strength and nonlinear plastic hardening described by the true stress power law sigma_t = k epsilon_t^n.

Explore ductile versus brittle failure in materials, comparing yield strength and fracture strength as failure criteria, and introducing leading failure theories like Tresca, maximum principal stress, and von Mises.

Explore distortion energy theory as the basis of yielding, decomposing strain energy into hydrostatic and deviatoric components, and derive the yield criteria for uniaxial and shear cases.

Explore von Mises yield criteria, representing yield surfaces for triaxial and biaxial states, and derive the von Mises equivalent stress from distortion energy, linking it to safety factor.

Explore the Tresca-Guest maximum shear stress theory, which defines failure as when the maximum shear stress exceeds the shear stress under uniaxial loading, with tau_max = (sigma1 - sigma3)/2 and the yield condition sigma1 - sigma3 >= sigma_y. Compare it with the von Mises (distortion energy) theory and the maximum normal stress theory, noting the common points of intersection on the uniaxial and 45-degree lines, the hexagonal bounding surfaces in two dimensions and the cylinder with hexagonal cross section in three dimensions, and that Tresca may yield a higher factor of safety or disagree with von Mises for brittle materials.

Explore fracture mechanics to account for cracks and other defects, and differentiate ductile fracture driven by shear from brittle fracture caused by normal stresses, noting crack paths and temperature effects.

Explore Griffith's postulate and fracture mechanics, balancing surface energy and elastic energy to predict critical crack length, while addressing stress concentration and plastic work g_c.

Explore linear elastic fracture mechanics, fracture modes one, two, three, and the relationship between the energy release rate and the stress intensity factor.

Examine how plastic zone size alters crack tip stresses and fracture behavior via K1C, with plastic deformation and plane stress in thin plates versus plane strain in thick plates.

Explain fracture toughness and strain energy release rate via crack opening displacement and the area under load-displacement curves, then use the R-curve to predict stable or unstable crack growth.

the j-integral provides a path-independent fracture criterion for small and large scale yielding, determinable from load-displacement curves, and in elastic cases equals G and relates to K_Ic via stress/strain relations.

Examine hardness testing as a practical, inexpensive quality-control method, focusing on Brinell hardness and indentation-based measures to assess material strength, heat treatment effects, and surface hardening.

Compare Rockwell, Vickers, and Knoop hardness tests—indenter types, loads, and unitless scales; Vickers yields clearer impressions, Knoop uses very low loads for soft materials and thin coatings.