Size effects in the bending of thin foils

The plastic response of a thin foil under bending is analysed using the phenomenological flow theory of strain gradient plasticity proposed by Fleck and Willis [N.A. Fleck, J.R. Willis, A mathematical basis for strain gradient plasticity theory. Part I: scalar plastic multiplier, J. Mech. Phys. Solids 57 (2009) 161–177; N.A. Fleck, J.R. Willis, A mathematical basis for strain gradient plasticity theory. Part II: tensorial plastic multiplier, J. Mech. Phys. Solids 57 (2009) 1045–1057]. Both the scalar and the tensorial versions of the theory are considered. Numerical results for elasto-plastic and elasto-viscoplastic foils are obtained via minimum principles, and closed-form expressions are derived in the limit of vanishing elasticity. An elevation of both bending moment at yield and hardening rate is predicted with decreasing foil thickness. Predictions are in line with existing experimental data on nickel foils with inferred material length-scales in the range 1–12 μm. Finally, the theory is used to assess size effects on the strength of metallic foams.