Question 1

What is the difference between the Ramberg-Osgood model and bilinear hardening in FEA?

Accepted Answer

Bilinear hardening uses two straight-line segments (elastic and plastic) with a constant hardening modulus. It is computationally cheap and converges reliably, making it suitable for large models at early design stages. The Ramberg-Osgood model uses a continuous nonlinear equation that accurately captures the smooth elastic-to-plastic transition, which is important for cyclic loading, fatigue analysis, and precise residual stress evaluation. In ANSYS this corresponds to Multilinear Isotropic Hardening (MISO); in ABAQUS to *PLASTIC with isotropic hardening.

Question 2

How are resilience and toughness calculated from the stress-strain curve?

Accepted Answer

Resilience (modulus of resilience) is the elastic strain energy density stored up to the yield point: U_r = σ_y² / (2E) [J/m³]. It represents the energy a material can absorb without permanent deformation. Toughness (modulus of toughness) is the total energy absorbed per unit volume up to fracture, calculated as the area under the entire stress-strain curve via numerical integration. Ductile metals have much higher toughness than brittle materials like cast iron or concrete.

Question 3

What is the tangent modulus and why does it matter in FEA?

Accepted Answer

The tangent modulus E_t = dσ/dε is the instantaneous slope of the stress-strain curve at a given strain. In the elastic region E_t equals Young's modulus E. Beyond yield, E_t decreases and represents the current stiffness of the material under continued plastic deformation. In FEA, the tangent modulus defines the stiffness matrix update in incremental plasticity algorithms (e.g., radial-return mapping). A low or zero tangent modulus can cause convergence difficulties and localization issues.

Question 4

How does the power-law hardening model (Hollomon equation) differ from Ramberg-Osgood?

Accepted Answer

The Hollomon power-law σ = Kε^n applies to the fully plastic regime and assumes the stress is purely a function of total strain (no separate elastic term). It is widely used for sheet metal forming analysis and necking prediction (Considère criterion: necking begins when n = ε). The Ramberg-Osgood model ε = σ/E + 0.002(σ/σ_y)^n includes both elastic and plastic strain contributions, making it more accurate near the yield point and standard in ASME fatigue assessment procedures.

Stress-Strain Curve & Material Nonlinear Models

Theory