Ductile cast iron (DCI) exhibits tension–compression asymmetry and pressure-sensitive yielding that conventional J2-plasticity cannot reproduce. Micromechanical models resolving graphite and interface effects have generally been assessed under uniaxial loading, whereas multiaxial studies have commonly idealized graphite nodules as voids. This work develops an experimentally validated micro-to-macro framework for a ferritic–pearlitic EN-GJS-600-3. A three-dimensional periodic Representative Volume Element (RVE) with statistically distributed graphite nodules combines radially oriented transversely isotropic graphite elasticity, unilateral matrix–graphite contact, and cooling-induced residual stresses. The matrix response is independently assessed through Vickers microindentation, while the homogenized RVE is validated against an experimental axial–torsional yield locus. The model incorporating graphite transverse isotropy and residual stresses reproduces this locus with an RMSE of 5.54 % and a maximum error of −9.27 %. The simulations show that interface opening under tension makes graphite nodules behave approximately as voids, whereas interface closure under compression transfers load to the nodules and increases the macroscopic yield strength. Cooling-induced compressive residual stresses further delay interface opening. Using an energy-based definition of yielding, general triaxial stress states are explored, and the yield surface is mapped in stress-invariant space. The predicted surface is closed, strongly pressure-sensitive and weakly dependent on the Lode angle; neither the Von Mises nor the conventional Drucker–Prager criterion reproduces this behavior. The micromechanical predictions are therefore condensed into a three-invariant Drucker–Prager Cap criterion suitable for component-scale Finite Element analyses. The proposed framework establishes a direct connection between graphite-scale mechanisms and a practical macroscopic constitutive description of DCI yielding.
We analyse the neutral-load spectrum of an Euler–Bernoulli beam with periodically alternating bending stiffness. A moment-based state formulation gives the exact transfer matrix of a unit cell and a scalar discriminant that controls the spectrum. For the infinite beam this discriminant separates the axial-load axis into buckling bands, which support bounded extended neutral modes, and gaps, which admit only spatially growing or decaying modes. A static Bragg condition proves that gaps open for every material contrast. This band-gap organisation is the static counterpart of pass bands and stop bands in periodic wave media, and it extends earlier studies of bi-periodic columns beyond the fundamental critical load. For a finite beam with pinned ends, Chebyshev polynomials give an exact characteristic equation that links the finite-column eigenloads to the infinite-medium spectrum. An exact first integral makes this equation necessary and sufficient. Each band carries one eigenload fewer than the number of cells, and each gap carries one end-localised mode, the static analogue of a surface state, whose load equals the critical load of the column braced at every cell junction. A long-wavelength expansion of the lowest eigenload recovers the harmonic-mean (Reuss) bending modulus and gives the first finite-cell correction. This correction decays rapidly with the number of cells and vanishes in the homogeneous limit, where Euler’s result is recovered exactly. The gaps belong to the linearised neutral problem, and post-buckling restabilisation lies outside their scope. The closed-form results provide benchmarks and a spectral basis for designing periodic load-bearing structures.
Microstructural anisotropy plays a central role in the mechanical response of granular materials and is commonly described by means of fabric tensors. This study investigates, by using a three-dimensional Discrete Element Method (DEM) numerical code, the evolution of three different fabric tensors, namely the normal contact-based, particle orientation-based, and void-based, with reference to both spherical and clumped particle assemblies, subject to triaxial loading–unloading.In small-strain elastoplastic constitutive modelling, common is the use of rotational hardening to simulate the anisotropic mechanical response of granular materials, in which the axis of both yield locus and plastic potential is called back stress, whose evolution is assumed to be related to the accumulation of irreversible strains.In this paper, with the aim of clarifying the relationship between back stress and different measures of microstructural fabric, the authors discuss the evolution, by using the stress probing technique, of fabric tensors with reversible/irreversible strains.The normal contact fabric is highly sensitive to minor, reversible rearrangements of microstructure and reaches stabilization prior to the critical state. The particle orientation fabric evolves primarily in response to significant irreversible strains and attains its asymptotic value only at critical state. During both loading and reversal unloading, the void-based fabric, taking mesoscale geometrical configuration into account, exhibits substantially lower sensitivity than the normal contact fabric to reversible strains, while its evolution is more closely associated with the development of irreversibility.
