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Based on the extended lower bound shakedown theorem for materials with internal rotation, variational principles for elastic and plastic shakedown are presented considering the indeterminate and modified couple-stress theory, proposed by Yang et al. (Int J Solids Struct 39(10):2731–2743, 2002), combined with the first strain gradient plasticity of Fleck and Hutchinson (J Mech Phys Solids 41(12):1825–1857, 1993). Using the concepts of convex analysis, a dual principle and the optimum conditions are derived. A numerical method based on the Newton’s method is derived from the plastic shakedown variational principle to find the load amplification factor. Two illustrative numerical examples are presented: a cantilever beam subjected to a constant axial load and to a variable external bending moment and a cylindrical bar subjected to an axial load and to a variable external torque. In both cases, the external torque and external bending moment vary arbitrarily between prescribed symmetric bounds. The results are compared with those obtained considering classical plastic shakedown theory in order to see the influence of the length scale on the load amplification factor. [ABSTRACT FROM AUTHOR]

Torque influences the main phenomena that occur during friction stir welding (FSW) process. However, models for torque have received little attention. In this paper, inverse problem method is used to estimate the parameters for a model for torque, measured during FSW experiments for different combinations of rotational and welding speeds. The experimental results are used as input data to estimate the model parameters. The results showed a good agreement between the experimental data and the model obtained using the inverse problem method. The influence of the tool geometry on torque was observed by comparing previously published experimental results and the experimental data presented. [ABSTRACT FROM AUTHOR]