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In this article, the Lippmann–Schwinger equation for nonlinear elasticity at small-strains is extended by mixed strain/stress gradient loadings. Such problems occur frequently, for instance when validating computational results with three-point bending tests, where the strain in the bending direction varies linearly over the thickness of the sample. To control all components of the effective strain/stress gradient the periodic boundary conditions are combined with constraints that enforce the periodically deformed boundary to approximate the kinematically fully prescribed boundary in an average sense. The resulting fixed point and Fletcher–Reeves algorithms preserve the positive characteristics of existing FFT-algorithms, like low memory consumption and extraordinary computational speed. The accuracy and power of the proposed methods is demonstrated with a series of numerical examples, including continuous fiber reinforced laminate materials.

Mixed strain/stress gradient loadings for FFT-based computational homogenization methods