Why non-linear?

• Linear (straight stress-strain line) or non-linear (curved stress-strain line)
• Elastic or plastic, due to material behaviour dependent on its molecular structuring & deformations characteristics

Hyperbolic model

• For non-linear Mohr-Coulomb strength approach

Tangent stiffness approach

• Stiffness at each point corresponds to the tangent of the stress-strain curve at point (continuous & differentiable)

Newton-Raphson correction

• Correct over-approximation of stiffness

Plasticity

• After elastic region, where strains & displacements are recoverable, reversible, elastic
• Plastic region occur, where permanent, irrecoverable, irreversible strains & displacements are produced

Yield surface

• The multi-stress surface where elastic regions end and plasticity starts

Plastic potential

• Surface where strain (e _v & e _q components) directional vector is normal to the yield surface

Flow

• Deformations of material in plasticity

Models

• Accounting for changes in material behaviour in regions of linear elasticity, non-linear elasticity, elastic-perfectly plasticity, elastic-plasticity

D-matrices

• Elastic: K = ò B^T.D_e.B dV; D_e: symmetric, positive-definite matrix
• Plastic: K_ep = ò B^T.D_ep.B dV
• D_ep: symmetric if associated flow (Y.S.=P.P.) and non-symmetric if non-associated

Integration

• Using Gaussian Quadrature integration method